graph

> This page is about the notion in [[combinatorics]]. For other notions of the same name see at _[[graph of a function]]_ and _[[graph of a functor]]_.

***

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### Context
#### Graph theory
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[[!include graph theory - contents]]
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# Contents
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##  Idea

A __graph__ is a collection of _[[vertices]]_ and _[[edges]]_; each [[edge]] links a pair of [[vertices]], defining a relationship of _incidence_ between vertices and edges.  There are several variations on the idea, described below.

This is the sense of graph in [[combinatorics]]; the other sense in high-school algebra, which interprets a [[morphism]] $f: A \to B$ as a [[subobject]] of the [[product]] $A \times B$, is unrelated; see _[[graph of a function]]_ for more on this.

##  Definitions

Let $V$ and $E$ be [[sets]]; call an element of $V$ a __vertex__ and an element of $E$ an __edge__.  A __graph__ is given by $V$, $E$, and a mapping $d$ that interprets edges as pairs of vertices.  Exactly what this means depends on how one defines 'mapping that interprets' and 'pair'; the possibilities are given below.  We will need the following notation:

*  $V^2$ is the [[cartesian product]] of $V$ with itself, the set of ordered pairs $(x,y)$ of vertices;
*  $\Delta_V$ is the [[diagonal subset]] of $V$, the set of pairs $(x,x)$, so that its [[complement]] $V^2 \setminus \Delta_V$ is the set of pairs as above where $x \neq y$;
*  $\left\langle{V \atop 2}\right\rangle$ is the [[quotient set]] of $V^2$ in which $(x,y)$ is identified with $(y,x)$, the set of unordered pairs $\{x,y\}$ of vertices;
*  $\left({V \atop 2}\right)$ is the quotient set of $V^2 \setminus \Delta_V$ in which $(x,y)$ is identified with $(y,x)$, the set of unordered pairs where $x \neq y$.

### Undirected graphs

We are now ready for the first batch of definitions.

*  For a __simple graph__, a pair of vertices is a [[subset]] of $V$ of [[cardinality]] $2$, and we interpret edges as unordered pairs of vertices in a one-to-one way.  Thus a simple graph is given by $V$, $E$, and an [[injective function]] $d: E \hookrightarrow \left({V \atop 2}\right)$.  Among graph theorists, this is often the default meaning of 'graph' unless another is specified. The category of simple graphs is called [[category of simple graphs|SimpGph]]

*  For a __[[multigraph]]__, a pair of vertices is the same as above, but we interpret edges as pairs of vertices in a many-to-one way.  Thus a multigraph is given by $V$, $E$, and an arbitrary [[function]] $d: E \to \left({V \atop 2}\right)$.

*  For a __loop graph__, a pair of vertices is any subset of the form $\{x,y\}$, where $x = y$ is allowed, and we interpret edges as pairs of vertices in a one-to-one way again.  Thus a loop-graph is given by $V$, $E$, and an injective function $d: E \hookrightarrow \left\langle{V \atop 2}\right\rangle$.

*  For a __[[pseudograph]]__, a pair of vertices is as in a loop graph, while edges are interpreted as pairs of vertices as in a multigraph.  Thus a pseudograph is given by $V$, $E$, and a function $d: E \to \left\langle{V \atop 2}\right\rangle$.

Note:  While 'simple graph' is unambiguous, the other terms above are not.  In particular, 'multigraph' sometimes means a pseudograph, 'pseudograph' sometimes means a loop graph, and 'loop graph' sometimes means a pseudograph.  These could be made unambiguous by saying 'simple multigraph', 'simple loop graph', and 'multipseudograph', respectively, but we will try to keep our terminology short.
An oldfashioned (e.g. [p. 142](#Harary1953)) term for 'simple graph' is 'linear graph', traces of which remain in the usual term 'linear hypergraph' in combinatorics (i.e. hypergraph any two edges of which intersect in at most one element of the ground set).

### Directed graphs

In all four of the above, edges are interpreted as *unordered* pairs.  If we instead interpret edges as *ordered* pairs, then we get four new concepts:

*  A __directed graph__ consists of $V$, $E$, and an injective function $d: E \hookrightarrow V^2 \setminus \Delta_V$;
*  a __directed multigraph__ consists of $V$, $E$, and a function $d: E \to V^2 \setminus \Delta_V$;
*  a __directed loop graph__ consists of $V$, $E$, and an injective function $d: E \hookrightarrow V^2$;
*  a __[[directed pseudograph]]__ consists of $V$, $E$, and a function $d: E \to V^2$.

Directed pseudographs are commonly used in [[category theory]], where they are often called 'directed graphs', 'digraphs', or (less ambiguously) '[[quivers]]'.

The same terminological ambiguities as above apply here as well, and they can be resolved in the same way, including using 'simple directed graph' for a directed graph if necessary.  One can also use 'undirected' in place of 'directed' to emphasise that the previous definitions apply instead of these.

It is always possible to interpret any kind of graph as a directed pseudograph (a quiver), in which there happens to be at most one edge between a given pair of vertices, or there happen to be no loops (or alternatively exactly one of every possible kind of loop), or in which there is an edge from $x$ to $y$ if and only if there is an edge from $y$ to $x$, or some mixture of these.

### Undirected graphs as directed graphs with an involution

There is an alternative definition of an undirected graph allowing loops and multiple edges (cf. [Serre 1977](#Serre1977)), that begins with the structure of a quiver $s,t : E \rightrightarrows V$ and then asks in addition for a [[fixed point]] free [[involution]] on edges $i : E \to E$ swapping source and target, i.e., such that
$$
i(i(e)) = e
\quad
i(e) \ne e
\quad
s(i(e)) = t(e)
\quad
t(i(e)) = s(e)
$$
Of course, since the source $s : E \to V$ and target $t : E \to V$ functions determine each other in the presence of the involution $i : E \to E$, it is enough to give, say, $s$ and $i$ to define an undirected graph.
In that case, one might alternatively view $E$ as a set of "half-edges" or "flags" (with $i$ the involution swapping the two halves of an edge), and even lift the condition that $i$ has no fixed points to allow for the possibility of undirected graphs with "dangling" or "open" edges (i.e., with only one side attached to a vertex).

Although this definition of undirected graphs with open edges is standard (cf. [[ribbon graph]]), [Kock (2016b)](#Kock2016BM) remarks that "it does not naturally lead to good notions of morphisms, beyond isomorphisms". A slight variation of this definition with a more natural notion of morphism was introduced by [Joyal and Kock (2009)](#JoyalKockQPL): they define a "Feynman graph" as a triple of [[finite sets]] $V, E, H$ together with a triple of a function $t : H \to V$, an injection $s : H \to E$, and a fixed point free involution $i : E \to E$. (See also [Kock (2016a)](#Kock2016GHP) for further discussion.)

### Orientations

An **orientation** of an undirected graph is the choice of a direction for every edge.
Formally, if we define undirected graphs as above to be quivers $E \rightrightarrows V$ equipped with a fixed point free involution $i : E \to E$, then an orientation corresponds to the choice of a subset $E^+ \subseteq E$ such that $E$ is the [[disjoint union]] $E = E^+ \uplus i(E^+)$.

Any orientation of an undirected graph induces a corresponding directed graph $E^+ \rightrightarrows V$.
In many situations, though, it is useful to consider a given undirected graph equipped with one of many possible orientations.
For example, various [[graph invariants]] (such as the [[flow polynomial]], or [[W. T. Tutte|Tutte]]'s original definition of the [[Tutte polynomial]]) can be defined for an arbitrary orientation of a graph, but are independent of the choice of orientation.

## Auxiliary definitions

The term __arc__ is often used for an *ordered* edge, while __line__ is sometimes used for an *unordered* edge.  We say that an arc $e$ with $d(e) = (x,y)$ is an arc __from $x$ to $y$__, while a line $e$ such that $d(e) = \{x,y\}$ is a line __between $x$ and $y$__.  In either case, a __loop__ is an edge from a vertex to itself or between a vertex and itself; only (possibly directed) loop graphs and pseudographs can have loops.

Given any sort of graph, we can define a [[binary relation]] on $V$; say that $x$ and $y$ are __adjacent__, written $x \sim y$, if there exists an edge $e$ such that $d(e) = (x,y)$ or $d(e) = \{x,y\}$.  A directed loop graph is determined entirely by this relation; we may say that it *is* $V$ equipped with a binary relation.  Then a simple directed graph is $V$ equipped with an [[irreflexive relation]] (or equivalently a [[reflexive relation]]), and an undirected loop graph is $V$ equipped with a [[symmetric relation]].

A graph is __[[finite graph|finite]]__ if $V$ and $E$ are both [[finite sets]].  Given a [[linear ordering]] of the [[vertices]] of a [[finite graph]], its __adjacency matrix__ is a square [[matrix]] whose $(i,j)$th entry gives the number of edges $e$ between the $i$th and $j$th vertices or from the $i$th vertex to the $j$th vertex.  In the examples above where a graph is determined by a binary relation on $V$, then matrix multiplication gives composition of relations.

##  Morphisms of graphs

An __[[isomorphism]]__ from $G = (V,E,d)$ to $G' = (V',E',d')$ consists of a bijection $f: V \to V'$, together with a bijection from $E$ to $E'$ (also denoted $f$) such that $f$ commutes with $d$; that is, $d(f(e)) = (f(x),f(y))$ or $d(f(e)) = \{f(x),f(y)\}$ whenever $d(e) = (x,y)$ or $d(e) = \{x,y\}$ (as appropriate).  Two graphs $G$ and $G'$ are __isomorphic__ if there exists such an isomorphism.  Then [[finite graphs]] $G$ and $G'$ are isomorphic if and only if they have the same number of vertices and, for some ordering of their vertices, they have the same adjacency matrix.

A __[[morphism]]__ from $G$ to $G'$ should consist of functions $f: V \to V'$ and $f: E \to E'$ such that $f$ commutes with $d$.  However, some authors allow $f(e)$ to be undefined if $d(e) = (x,y)$ or $d(e) = \{x,y\}$ but $f(x) = f(y)$ when using a notion of graph where loops are forbidden.  The difference amounts to whether one interprets a simple graph as a special kind of loop graph in which no loops exist (the first kind of morphism) or in which each vertex has a unique loop (the second kind of morphism).  Either way, an isomorphism (as defined above) is precisely an invertible morphism.

Under the second notion of morphism (where simple graphs are identified with sets equipped with a symmetric reflexive relation), the [[category of simple graphs]] has many desirable properties (q.v.).

## Notions of subgraphs from the nPOV

A usual definition of _subgraph_ in combinatorics is, roughly: _subset_.
More precisely, if _undirected simple graph_ means _pair $(V,E)$ of two sets, with $E\subseteq[V]^2$ any subset of the set of all two-element subsets of $V$_, then a usual meaning of _subgraph_ of $(V,E)$ is (cf. e.g. [p. 3](#DiestelGraphTheoryFourthEd)): any pair $(W,F)$ with $W\subseteq V$, $F\subseteq E$, and[^1] $F\subseteq [W]^2$.

[^1]: The last condition is to ensure that $(W,F)$ is itself again a graph, which would not be guaranteed if we defined subgraph to be just any pair of subsets of the respective sets $V$ and $E$.

In particular, such a subgraph $(W,F)$ may have isolated vertices which are not isolated in the ambient graph $G$, and $(W,F)$ need not be an _induced_ subgraph, which by definition is a subgraph in the above sense for which $F = [W]^2\cap E$. In words: the edge-set of an _induced_ subgraph must contain all edges _induced_ within the ambient graph by the vertex set of the subgraph.[^2]

[^2]: This happens to be the usual notion of substructure in model theory, for any relational structure.

Another usual notion of subgraph in combinatorics is[^3] _spanning_ subgraph:

[^3]: Somewhat counterintuitively (with regard to connotations of the words _spanning_ and _induced_), a _spanning_ subgraph need not be _induced_, and in fact it never is, _except_ if the subgraph is the graph itself.

this means just any subgraph $(W,F)$ in the above sense with $W=V$. There are three kinds of spanning subgraphs which are the most studied: [[Hamilton circuit|Hamilton circuits]][^4], [[perfect matching|perfect matchings]] and [[spanning tree|spanning trees]].

[^4]: We here follow A. Bondy's choice of words in [p. 20](#HandbookOfCombinatoricsVol1), both in the decision whether to use _hamiltonian_ or _Hamilton_, and whether to use _cycle_ or _circuit_.
The term _circuit_ is less usual than _cycle_ in combinatorics, but less ambiguous, not longer, and more clearly signalling that the combinatorial notion is meant (not one of the many other meanings of _cycle_).
One argument in favor of _Hamilton_ is that _any_ circuit, by itself, is _hamiltonian_.

From the [[nPOV]], it is often possible to describe notions of subgraph in terms of types of monomorphisms in categories of graphs; for example,

* _subgraphs_ are [[monos]], and

* _induced subgraphs_ are [[regular monos]]

in the [[category of simple graphs]], and similarly for suitable categories of other types of graph.

One synonym, in the nLab[^5] for _induced subgraph_ is *full subgraph*, for brevity, and for harmony with other uses of _full_ in category theory (but also for more precise reasons).

[^5]: Incidentally, the term _full_ was in use in mid-twentieth century graph theory, then seems to have fallen out of favor.

The precise meaning of _subgraph_ depends on the chosen formalization of *graph*, needless to say.

## Undirected graphs as 1-complexes, barycentric subdivision

Recall that a [[simplicial complex]] of dimension one consists of the data of a set $V$ together with a set $S$ of non-empty subsets of $V$ of cardinality at most $2$, that contains all of the [[singleton]] subsets.
In other words, a 1-dimensional simplicial complex is essentially the same thing as a simple graph, with the set of edges being determined by the set of simplices and vice versa:
$$
E = \{\{x,y\} \mid \{x,y\} \in S, x \ne y\}
\qquad
S = \{\{x\} \mid x \in V\} \cup E
$$
For this reason, simple graphs are sometimes referred to as **simplicial graphs** [(Gross & Tucker 1987)](#GrossTucker).
On the other hand, an undirected graph $G$ with loops or multiple edges can more generally be seen as a 1-dimensional [[CW-complex]] (or more precisely, it has a [[geometric realization]] $|G|$ as a CW-complex in which 0-cells correspond to vertices and 1-cells to edges).

Given a general undirected graph, it is always possible to obtain a simple graph through the process of _[[barycentric subdivision]]_.
Let $G$ be a graph with vertex set $V$ and edge set $E$.
The **barycentric subdivision** of $G$ is the graph $G'$ with vertex set $V \cup E$, and with an edge joining $v \in V$ to $e \in E$ just in case $v$ is incident to (i.e., at either end of) $e$ in $G$.
It is easy to check that $G'$ contains no loops, while the two-fold barycentric subdivision $G''$ contains no loops or multiple edges, in other words is a simple graph.
(More generally, the $n$-fold barycentric subdivision contains no circuit of length $\le n$).
Part of the reason for the importance of simple graphs is that many "topological" properties of a graph $G$ (such as [[planar graph|planarity]], first [[Betti number]], etc., which can be defined in terms of the geometric realization of $G$) are preserved under barycentric subdivision.
(Although obviously, not all _graph-theoretic_ properties are preserved. For example, barycentric subdivision always produces a [[bipartite graph]]).

## Flavors of graphs

* [[reflexive graph]]

* [[directed graph]]

## Lawvere's remarks on graph theory

At the [[Como]] conference in 1990, [[William Lawvere]] gave a videotaped lecture including the following remarks:

> {#LawvreOnGraphTheory} I have great problems reading books on graph theory, books and papers on graph theory, because they never tell you exactly what they are talking about. Sometimes the graphs are [word inaudible, even when played slower], sometimes they are absolutely reflexive, sometimes they are not.
Even if they go so far as talking about homomorphisms, I still don't know exactly what that is, i.e., which category are we in?
What they should do is admit that they are working in three or four *different* categories and they don't know how to pass from one to the other, and so on, and [inaudible words] to simplify.
But no, they prefer to talk in a vague way and smushing these together. [inaudible] tried to understand some of the problems of graph theorists and get [bogged? locked?] in the first page.
Does anybody actually know what a graph minor is?
[some interjection from the audience] Graph minor. Big problem.
(..) you see, this famous [inaudible works] problem on graph minors.
Looks like that that might be interesting.
But I can't determine *exactly* what it is, because, if you read the first parts of the paper, they waffle, you see, they don't give you a *property* (...)
([William Lawvere] in his 1990 lecture at [[Como]], Italy, Villa Olmo)

## In constructive mathematics

The notion of graph bifurcates in [[constructive mathematics]]:

* The set of edges of a graph could be defined with a [[denial inequality]]:
$$E \coloneqq \{(x, y) \in V \times V \vert x \neq y\}$$

* The set of edges of a graph could be defined with a [[tight apartness relation]]:
$$E \coloneqq \{(x, y) \in V \times V \vert x \# y\}$$

## Related concepts

* [[omega-graph]]

* [[planar graph]], [[connected graph]]

* [[acyclic graph]]

* [[signed graph]]

* [[empty graph]]

* [[ribbon graph]]

* [[hypergraph]]

* [[quiver]], [[McKay quiver]]

* [[n-quiver]]

* [[distance (graph theory)]]

* [[relation]]

* [[category]], [[free category]]

* [[Feynman diagram]]

* [[graph complex]], [[formality of the little n-disk operad]]

* [[graph of groups]]

* [[semi-graph]]

* [[loop graph object]]

## References

Textbooks:

* Frank Harary (1969), _Graph Theory_, Addison-Wesley.

* Frank Harary and E.M. Palmer (1973), _Graphical Enumeration_, Academic Press.

* {#Serre1977} [[Jean-Pierre Serre]] (1977), _Trees_, Springer.

* [[W. T. Tutte]] (1984), _Graph Theory_, Addison-Wesley.

* {#GrossTucker} Jonathan L. Gross and Thomas W. Tucker (1987), _Topological Graph Theory_, Wiley.

* Gunther Schmidt and Thomas Ströhlein (1993), _Relations and Graphs: Discrete Mathematics for Computer Scientists_, EATCS Monographs on Theoretical Computer Science, Springer.

* {#HandbookOfCombinatoricsVol1} A. Bondy. Basic Graph Theory: Paths and Circuits. Elsevier Amsterdam, 1995, Vol. 1, pp. 1--110

* Chris Godsil and Gordon Royle (2001), _Algebraic Graph Theory_, Springer.

* {#DiestelGraphTheoryFourthEd} [[Reinhard Diestel]], *Graph Theory*, Graduate Texts in Mathematics **173** 5th edition (2017) [[website](https://diestel-graph-theory.com/), [doi:10.1007/978-3-662-53622-3](https://doi.org/10.1007/978-3-662-53622-3)]

Other references:

* [[Bill Lawvere]] (1989), _Qualitative distinctions between some toposes of generalized graphs_, in Categories in computer science and logic (Boulder, CO,   1987), volume 92 of _Contemporary Mathematics_, 261--299. American Mathematical Society, Providence, RI.

* E. Babson, H. Barcelo, M. de Longueville, R. Laubenbacher, A Homotopy Theory for Graphs, [arXiv:math/0403146 ](http://arxiv.org/abs/math/0403146)

* [[Ronnie Brown]], I. Morris, J. Shrimpton, and C.D. Wensley (2008), _Graphs of Morphisms of Graphs_, Electronic Journal of Combinatorics, A1 of Volume 15(1), 1--28.

* {#Harary1953} [[Frank Harary]]: _On the notion of balance of a signed graph_. The   Michigan Mathemathical Journal, Volume 2, Issue 2 (1953), 143-146.

* {#JoyalKockQPL} [[André Joyal]] and [[Joachim Kock]], Feynman graphs, and nerve theorem for compact symmetric multicategories (extended abstract), in _Proceedings of the 6th International Workshop on Quantum Physics and Logic_(Oxford 2009), Electronic Notes in Theoretical Computer Science 270 (2) (2011), 105-113. [arXiv:0908.2675](https://arxiv.org/abs/0908.2675)

* {#Kock2016GHP} [[Joachim Kock]], Graphs, hypergraphs, and properads, _Collect. Math._ 67 (2016), 155-190. [arXiv:1407.3744](https://arxiv.org/abs/1407.3744)

* {#Kock2016BM} [[Joachim Kock]], Cospan construction of the graph category of Borisov and Manin, [arXiv:1611.10342](https://arxiv.org/abs/1611.10342)

* Martin Schmidt, _Functorial Approach to Graph and Hypergraph Theory_, ([arXiv:1907.02574](https://arxiv.org/abs/1907.02574))

See also [[quiver#references|quiver - references]].

Discussion of use of [[simplicial complexes]] in graph theory:

* MO, _[What have simplicial complexes ever done for graph theory?](http://mathoverflow.net/questions/161586/what-have-simplicial-complexes-ever-done-for-graph-theory)_

[[!redirects linear graph]]
[[!redirects graph]]
[[!redirects graphs]]
[[!redirects undirected graph]]
[[!redirects undirected graphs]]
[[!redirects simple directed graph]]
[[!redirects undirected simple graph]]
[[!redirects simple directed graphs]]
[[!redirects graph theory]]


graph complex

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#### Cohomology
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#Contents#
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## Idea
 {#Idea}

A _graph complex_ is a certain [[cochain complex]] [[linear span|spanned]] by [[equivalence classes]] of certain labeled [[directed graphs]], whose [[differential]] encodes the operation of contracting away [[edges]] in a graph.

Two similar but different classes of examples are usually referred to by default as just "the graph complex", going back to hints in [Kontsevich 92, P. 11-12](#Kontsevich92), [Kontsevich 93, 5](#Kontsevich93):

Given a [[smooth manifold]] $\Sigma$, often taken to be [[Euclidean space]] $\mathbb{R}^D$, there is

1. a [graph complex model for the real cohomology of the unlabeled ordered configuration space of points](#IdeaGraphComplexModelsForConfigurationSpacesOfPoints) in $\Sigma$

1. a [graph complex model for the real cohomology of the space of knots](#IdeaGraphComplexModelForSpacesOfKnots) in $\Sigma$.

In both cases, the graphs are interpreted as [[Feynman diagrams]] for [[Chern-Simons theory]] and the map which identifies these with [[cocycles]] in [[real cohomology]] of either the (unlabeled & ordered) [[configuration space of points]] or the [[space of knots]] is given by sending a Feynman diagram to its [[Feynman amplitude]]. In the first case this is an [[n-point function]], [[correlator as differential form on configuration space of points|regarded as a differential form on the configuration space]] of $n$-points, while in the second case this is a [[vacuum amplitude]] depending on the [[isotopy]] [[equivalence class|class]] of the  [[Wilson loop]] encoded by the [[knot]] -- a [[Vassiliev knot invariant]].

\linebreak

**Beware** that these graph complex models **differ only in very small technical detail** as the manifold $\Sigma$ varies and the choice between models for configuration space of points or spaces of knots is made (see the [Overview of definitions](#OverviewOfDefinitions) below), and yet these small details **completely change the cohomological nature** of the resulting graph complexes; a point not often made manifest when any given author discusses  "the graph complex".

\linebreak

### Model for ordered configuration spaces of points
 {#IdeaGraphComplexModelsForConfigurationSpacesOfPoints}

The graph complex model for unlabeled & ordered [[configuration spaces of points]] was originally sketched in [Kontsevich 92 (p. 11-12)](#Kontsevich92) and worked out in detail in [Lambrechts-Volić 14](#LambrechtsVolic14) for $\Sigma = \mathbb{R}^D$ a [[Euclidean space]]. Other authors have claimed generalization to $\Sigma$ a [[closed manifold]] ([Campos-Willwacher 16](#CamposWillwacher16)) possibly [[manifold with boundary|with boundary]] ([Campos-Idrissi-Lambrechts-Willwacher 18](#CamposIdrissiLambrechtsWillwacher18)).

Here we denote this version of the graph complex by "$Graphs$", in contrast to "$KnotGraphs$" for the other model, discussed further [below](#IdeaGraphComplexModelForSpacesOfKnots).

This graph complex is [[linear span|spanned]] by [[finite set|finite]] [[directed graphs]] with [[linear order|linearly ordered]] internal and external [[vertices]], subject to a sign rule reflecting the [[orientation]] of [[edges]]. The [[differential]] sends any graph to a signed sum of graphs obtained by contracting one of the [[edges]] with at least one internal vertex.

Under ordered disjoint union of edges and _internal_ vertices this [[cochain complex]] becomes a [[differential graded-commutative algebra]]

$$
  Graphs_n\big( \mathbb{R}^D \big)
  \;\in\;
  dgcAlg_{\mathbb{R}}
  \phantom{AA}
  n, D \in \mathbb{N}
  \,,
$$

As such, this is [[quasi-isomorphism|quasi-isomorphic]] to the semi-algebraic [[de Rham algebra]]

$$
  \Omega^\bullet
  \big(
    \underset{{}^{1,\cdots,n}}{Conf}
    \big(
      \Sigma
    \big)
  \big)
  \;\in\;
  dgcAlg_{\mathbb{R}}
  \phantom{AA}
  n, D \in \mathbb{N}
$$

of the [[Fulton-MacPherson compactification]] of the [[configuration space of points]] $\underset{{}^{1,\cdots,n}}{Conf}(\mathbb{R}^D)$ for $n$ points in $D$-dimensional [[Euclidean space]].

The [[chain map]] which exhibits this [[quasi-isomorphism]] is given by regarding a graph as a [[Feynman diagram]] for [[Chern-Simons theory]] on $\Sigma = \mathbb{R}^D$ and sending it to its corresponding [[Feynman amplitude]], namely to the [[configuration space of points|configuration space]]-[[integral]] of the [[wedge product]] of [[Chern-Simons propagators]] associated to the [[edges]], regarding [[Feynman amplitudes as differential forms on configuration spaces of points]]:

\[
  \label{TheQuasiIsomorphism}
  \underset{
    \color{blue}
    \array{
      \phantom{A}
      \\
      \text{graph complex}
      \\
      \text{of n-point Feynman diagrams}
      \\
      \text{for Chern-Simons theory}
      \\
      \text{on} \; \Sigma
    }
  }{
    Graphs_n(\Sigma)
  }
  \underoverset{
    \simeq_{\mathrlap{qi}}
  }
  {
    \color{blue}
    \array{
      \text{assign Feynman amplitudes}
      \\
      \text{of Chern-Simons theory}
      \\
      \phantom{A}
    }
  }
  {
    \longrightarrow
  }
  \underset{
    \color{blue}
    \array{
      \phantom{A}
      \\
      \text{de Rham algebra of}
      \\
      \text{ordered configuration space}
      \\
      \text{of n points in}\; \Sigma
    }
  }{
  \Omega^\bullet
  \big(
    \underset{{}^{ \{1,\cdots,n \} }}{Conf}\big(  \Sigma \big)
  \big)
  }
  \,.
\]

This means that for each [[edge]] in a [[graph]] a [[Chern-Simons propagator]] is assigned, and for each of $n_{int} \in \mathbb{N}$ internal [[vertices]] the [[wedge product]] of its adjacent [[Chern-Simons propagators]] is [[fiber integral|fiber integrated]] along the canonical [[fibration]] of [[configuration spaces of points]]:

$$
  \underset{
    {}^{\{1, \cdots, n + n_{int} \}}
  }{
    Conf
  }
  \big(
    \mathbb{R}^D
  \big)
  \longrightarrow
  \underset{
    {}^{\{1,\cdots,n\}}
  }{
    Conf
  }
  \big(
    \mathbb{R}^D
  \big)
$$

just as befits the definition of a [[Feynman amplitude]] when [[correlator as differential form on configuration space of points|regarding them as differential forms on configuration spaces of points]].

\linebreak

### Model for spaces of knots
 {#IdeaGraphComplexModelForSpacesOfKnots}

The graph complex model for [[spaces of knots]] was originally sketched in [Kontsevich 93, Section 5](#Kontsevich93) and worked out in [Cattaneo, Cotta-Ramusino, Longoni 02](#CattaneoCottaRamusinoLongoni02), [05](#CattaneoCottaRamusinoLongoni05), see also [Bar-Natan 91](#BarNatan91).

Here we denote this version of the graph complex by "$KnotGraphs$", in contrast to "$Graphs$" for the other model, discussed [above](#IdeaGraphComplexModelsForConfigurationSpacesOfPoints).

The explicit definition is almost exactly the same as that of the model for configuration spaces of points [above](#IdeaGraphComplexModelsForConfigurationSpacesOfPoints), except that the external vertices here have a degree -1 instead of 0, and that the differential sees contractible [[edges]] between consecutive external vertices, often called _arcs_, shown by dashed lines on the right.

Together these two innocent modifications make the graphs now represent [[vacuum amplitude|vacuum]] [[Feynman diagrams]] for [[Chern-Simons theory]] in the presence of a [[Wilson loop]], with the external vertices now attached to this [[knot]] (corresponding to the dashed line shown on the right):

\[
  \label{KnotGraphsTheQuasiIsomorphism}
  \underset{
    \color{blue}
    \array{
      \phantom{A}
      \\
      \text{graph complex}
      \\
      \text{of n-point Feynman diagrams}
      \\
      \text{for Chern-Simons theory}
      \\
      \text{on} \; \Sigma
      \\
      \text{in the presence of a Wilson loop knot}
    }
  }{
    KnotGraphs_n(\Sigma)
  }
  \underoverset{
  }
  {
    \color{blue}
    \array{
      \text{assign Feynman amplitudes}
      \\
      \text{of Chern-Simons theory}
      \\
      \phantom{A}
    }
  }
  {
    \longrightarrow
  }
  \underset{
    \color{blue}
    \array{
      \phantom{A}
      \\
      \text{de Rham algebra}
      \\
      \text{of the space of knots}
      \\
      \text{in}\; \Sigma
      \\
      \text{(higher Vassiliev knot invariants)}
    }
  }{
  \Omega^\bullet
  \big(
    Emb(S^1,  \Sigma \big)
  \big)
  }
  \,.
\]

This yields higher [[Vassiliev knot invariants]], a good review is in [Volić 13](#Volic13). It is a [[conjecture]] ([Volić 13, Conjecture 4.8](#Volic13)) that this map, too, is a [[quasi-isomorphism]].

\linebreak

### Model for other spaces

There are yet other, inequivalent, graph complexes. Notably there is a type of graph complex whose (co)homology in degree 0 is the [[Lie algebra]] of the [[Grothendieck-Teichmüller group]] ([Willwacher 10](#Willwacher10), [Dolgushev-Rogers 12](#DolgushevRogers12)), and this is neither of the above two cases. (...)

## Definition
 {#GraphComplex}

### Overview
 {#OverviewOfDefinitions}

There are two different classes of (Kontsevich-) graph complexes, modelling the [[real cohomology]] of, respectively,

| [[configuration space]]   |    |   |  [[graph complex]]  |  |
|----|----|---|----|--|
| [[configuration spaces of points]] | $\underset{{}^{1,\cdots,n}}{Conf}(\Sigma)$ | $\;\;\leftrightarrow\;\;$ | $Graphs_n(\Sigma)$ |  [(1)](#GraphComplexesModellingConfigurationSpacesOfPoints) |
| [[spaces of knots]] | $Emb(S^1,\Sigma)$ | $\;\;\leftrightarrow\;\;$ | $KnotGraphs(\Sigma)$ | [(2)](#GraphComplexesModellingSpacesOfKnots) |

Despite their very deifferent cohomological nature, the explicit definitions of these two kinds of graph complexes are mostly identical, except for some small but crucial difference in the definition of degrees and labels of edges (see [below](#KnotGraphsDefinition)).

In addition, the definition of the graph complex model for configuration spaces of points depends on a choice of [[smooth manifold]] $\Sigma$, possibly [[manifold with boundary|with boundary]], namely such that the graph complex $Graphs(\Sigma)$ provides a model for the cohomology of the [[configuration space of points]] in $\Sigma$

$$
  Graphs\big( \Sigma \big)
  \leftrightarrow
  \Omega^\bullet\big(  Conf(\Sigma) \big)
  \,.
$$

The [[configuration spaces of points]] are naturally [[filtering|filtered]] by [[subspaces]] of a fixed number $n \in \mathbb{N}$ of points. In the graph complexes this correpsonds to graphs with a fixed number $n$ of _external_ vertices:

$$
  \underset{
    \mathclap{
    \color{blue}
    \array{
       \phantom{A}
       \\
       \text{graphs with}
       \\
        n\;\text{external vertices}
    }
    }
  }{
    Graphs_n\big( \Sigma\big)
  }
  \;\;\;\leftrightarrow\;\;\;
  \Omega^\bullet\big(
     \underset{
       \mathclap{
       \color{blue}
       \array{
         \phantom{A}
         \\
         \text{configuration space}
         \\
         \text{of}\;n\;\text{points}
       }
       }
     }{
       \underbrace{
         \underset{{}^{1,\cdots,n}}{Conf}(\Sigma)
       }
     }
  \big)
  \,.
$$

The full graph complexes/configuration spaces are just the [[direct sum]]/[[union]] of those for fixed number $n$ of external vertices/points, hence one may focus attention on the latter.

The precise form of these relations is the content of the theorems discussed below. Before stating this in detail, we make some general remarks on how the situation depends on $\Sigma$:

Beware that the graphs $[\Gamma] \in Graphs_n(\Sigma)$ themselves are _not_ going to carry an embedding into the manifold $\Sigma$, they are just abstract graphs. But the construction of the above correspondence to the cohomology of the configuration space of points is given by associating with a graph the corresponding [[correlator as differential form on configuration space of points|correlator]] in a [[Chern-Simons theory|Chern-Simons]] [[perturbative quantum field theory]] on the space $\Sigma$.

As a consequence, the dependence of the graph complexes themselves on $\Sigma$ is mild:

In particular, in the case that $\Sigma = \mathbb{R}^D$ is a [[Euclidean space]], the corresponding graph complexes $Graphs\big( \mathbb{R}^D \big)$ depend essentially only on whether the [[dimension]] $D$ is [[even number|even]] or [[odd number|odd]]. Concretely, the degree of a graph in the [[graded vector space]] $Graphs_n(\mathbb{R}^D)$ is

\[
  \label{DegreeOfAGraph}
  deg\big( [\Gamma]\big)
  \;=\;
  (D-1) \cdot \#\!Edges - D \cdot \#\!Vertices_{int}
  \,,
\]

hence each [[edge]] contributes a degree $D-1$ and each internal vertex a degree $-D$. As a consequence,  due to the [[signs in supergeometry|sign rule]] in the [[graded-commutative algebra]] [[structure]] on $Graphs_n(\mathbb{R}^D)$ we have the following parities

|   $\Sigma = \mathbb{R}^D$ | edges | internal vertices |
|-------------------|-------|-------------------|
| $D$ even |  odd  | even |
| $D$ odd  | even | odd |

Up to an absolute _even_ change of grading, the graph complexes $Graphs_n(\mathbb{R}^D)$ depend only on these parities, hence depend only on whether $D$ is [[even number|even]] or [[odd number|odd]].

Finally, beware that many authors consider the case where $\Sigma = \mathbb{R}^D$ by default, and don't mention a dependence on a choice of manifold, but just on a natural number $D$. This natural number is then often denoted "$d$" or "$n$".

For instance:

* the "graph complex with anti-symmetric set of edges" of [Bar-Natan & McKay, Def. 3.3](#BarNatanMcKay) is denoted $GC_2$ in [Willwacher 10](#Willwacher10) and is our $Graphs_0\big(\mathbb{R}^2\big)$;

* similarly $GC_3$ in [Koroshkin-Willwacher-Živković 14](#KoroshkinWillwacherZivkovic14) is our $KnotGraphs\big( \mathbb{R}^3\big)$.

(While the cohomology of $Graphs_0(\Sigma)$ -- being equivalent to that of $Conf_0(\Sigma) \simeq \ast$ -- is trivial, these authors consider further filtrations which have non-trivial cohomology in filtration stages, see [Bar-Natan & McKay, Def. 3.6](#BarNatanMcKay)).

\linebreak

### **1)** Graph complexes modelling configuration spaces of points
 {#GraphComplexesModellingConfigurationSpacesOfPoints}

We discuss graph complexes $Graphs_n(\Sigma)$ modelling the [[real cohomology]] of [[configuration spaces of points]] $\underset{{}^{1,\cdots,n}}{Conf}(\Sigma)$.

For the case of $\Sigma = \mathbb{R}^D$ these were originally hinted at in [Kontsevich 92 (p. 11-12)](#Kontsevich92) and discussed in detail in [Lambrechts-Volić 14](#LambrechtsVolic14).

\linebreak

#### $Graphs_n\big( \mathbb{R}^3 \big)$
{#GraphComplexForConfigurationsOfPointsInR3}

We state the definition of the graph complexes $Graphs_n\big( \mathbb{R}^3\big)$ (as motivated [above](#IdeaGraphComplexModelsForConfigurationSpacesOfPoints), see Def. \ref{GraphComplexDgcAlgebra} below) associated with [[Euclidean space]] of [[dimension]] 3, following [Lambrechts-Volić 14, Section 6](#LambrechtsVolic14).

##### Graphs

+-- {: .num_defn #Graphs}
###### Definition
**(graphs)**

In the following, by a **[[graph]]** we mean specifically:

1. a [[finite set|finite]] [[directed graph]]

   $$
     Edg \underoverset{t}{s}{\rightrightarrows} Vert
     \,,
   $$

1. a decomposition  of the [[finite set]] $Vert$ of [[vertices]] as a [[disjoint union]]

   $$
     Vert \simeq Vert_{ext} \sqcup Vert_{int}
   $$

   of "external" and "internal" vertices.

1. a [[linear order]] on $Vert$, such that $Vert_{ext} \lt Vert_{int}$.

Hence both $Vert_{ext}$ and $Vert_{int}$ are [[linear orders]], and

$$
  Vert = Vert_{ext} \sqcup_{{}_{\lt}} Vert_{int}
$$

is their ordered [[disjoint union]].

An [[isomorphism]] $\Gamma \simeq \Gamma'$ of such graphs is a [[pair]] of [[bijections]] between sets of edges and vertices

\[
  \label{IsomorphismOfGraphs}
  \array{
    Edg &\underoverset{t}{s}{\rightrightarrows}& Vert_{ext} \sqcup Vert_{int}
    \\
    \mathllap{{}^{\simeq}}\big\downarrow
    &&
    \mathllap{{}^{\simeq}}\big\downarrow
    \\
    Edg' &\underoverset{t'}{s'}{\rightrightarrows}& Vert'_{ext} \sqcup Vert'_{int}
  }
\]

which respects all [[structure]], hence the [[source]] and [[target]] maps, the decomposition into external and internal vertices, and the [[linear order]].

=--

([Lambrechts-Volić 14, Def. 6.1](#LambrechtsVolic14))

+-- {: .num_remark #EmptyGraph}
###### Remark
**([[empty graph]])**

The [[empty graph]] counts as a graph, according to Def. \ref{Graphs}.

=--

We write $\#\! Vert \coloneqq \left\vert Vert \right\vert$ for the [[cardinality]] of the sets of vertices, etc.

+-- {: .num_defn #DenotationForLinearOrderOnVertices}
###### Remark
**(denotation for linear order on vertices)**

The  [[linear order]] on the [[vertices]] in Def. \ref{Graphs} is equivalently a choice of [[bijection]] of $Vert$ with the set of the first $\#\!Vert$ [[natural numbers]]

\[
  \label{NaturalNumberLabelsOfVertices}
  Vert
  \;\simeq\;
  \big\{
    1, 2, \ldots, \# Vert_{ext}, \# Vert_{ext} + 1, \ldots , \# Vert_{ext} + \# Vert_{int}
  \big\}
\]

With this understood, we may depict graphs as as diagrams with oriented edges and numbered vertices.

=--

+-- {: .num_defn #DegreeOfAGraph}
###### Definition
**(degree of a graph)**

Given a graph

$$
  \Gamma
  \;=\;
  \big(
    Edg
      \underoverset{t}{s}{\rightrightarrows}
    Vert_{ext}
      \sqcup
    Vert_{int}
  \big)
$$

according to Def. \ref{Graphs}, its _degree_ is the [[natural number]]

$$
  deg(\Gamma)
  \;\coloneqq\;
  2 \cdot \#\!Edg
  -
  3 \cdot \#\!Vert_{int}
  \,,
$$

hence the sum of 2 for each [[edge]] and of -3 for each internal [[vertex]].

=--

([Lambrechts-Volić 14, Def. 6.6](#LambrechtsVolic14))

If we also agree to

1. denote internal vertices by a bullet;

1. denote external vertices by... _no_ denotation (as usual for [[Feynman diagrams]]!)

then the ingredients of these graphs are as shown in the following figure:

The following shows some very simple examples of graphs, depicted this way:

**Beware** that graphs need not be [[planar graph|planar]], may have edges between two external vertices (chords) and other effects not seen in these simple examples (... need to draw more generic examples...).

In particular, graphs may have [[tadpoles]]. But the next definition makes such graphs vanish in the graph complex:

+-- {: .num_defn #SignRulesForGraphs}
###### Definition
**(sign rules for graphs)**

For a fixed [[finite set|finite]] [[linear order]] $Vert_{ext} \simeq \{1, 2, \cdots, \#\!Vert_{ext}\}$, consider the [[real numbers|real]] [[linear span]] on the [[set]] of [[isomorphism classes]] (eq:IsomorphismOfGraphs) of graphs, according to Def. \ref{Graphs}, with that set of external [[vertices]]:

\[
  \label{LinearSpanOfIsoClassesOfGraphs}
  \mathbb{R}
  \Big[
    \big\{
     Edg
      \underoverset{t}{s}{\rightrightarrows}
     Vert_{ext}
      \sqcup
     Vert_{int}
     \;\vert\;
     Vert_{int} \in FinLinOrd
    \big\}_{/\sim}
  \Big]
  \;\in\;
  Vect_{\mathbb{R}}
  \,.
\]

The degree associated with a graph (Def. \ref{DegreeOfAGraph}) makes this a [[graded vector space]].

On this [[real vector space]], consider the linear [[equivalence relation]]  $\sim$ generated by

\[
  \label{SignFromOrientationReversal}
  \Gamma
  \;\sim\;
  -\Gamma'
  \phantom{AA}
  \array{
    \text{if}\;\Gamma'\; \text{is}\;\Gamma\;\text{with}
    \\
    \text{orientation of one edge reversed}
  }
\]

and

\[
  \label{SignRuleFromOrderingInternalVertices}
  \Gamma
  \;\sim\;
  -\Gamma'
  \phantom{AA}
  \array{
    \text{if}\;\Gamma'\;\text{is}\;\Gamma\;\text{with}
    \\
    \text{a pair of consecutive internal vertices transposed}
  }
\]

=--

([Lambrechts-Volić 14, Def. 6.5 & Def. 6.5](#LambrechtsVolic14))

Since this [[equivalence relation]] is linear, the set of its [[equivalence classes]] is still a [[vector space]], and since the equivalence relation also preserves the degrees (Def. \ref{DegreeOfAGraph}), it is still a [[graded vector space]]:

+-- {: .num_defn #GradedVectorSpaceOfGraphs}
###### Definition
**(graded vector space of graphs)**

For $n \in \mathbb{N}$, $n \coloneqq \#\! Vert_{ext}$, we write

\[
  \label{VectorSpaceOfGraphsWithSignRulesImposed}
  \widehat{Graphs}_n\big(\mathbb{R}^3\big)
  \;\coloneqq\;
  \mathbb{R}
  \Big[
    \big\{
     Edg
      \underoverset{s}{t}{\rightrightarrows}
     Vert_{ext}
      \sqcup
     Vert_{int}
     \;\vert\;
     Vert_{int} \in FinLinOrd
    \big\}_{/\sim}
  \Big]_{/\sim}
  \;\in\;
  Vect_{\mathbb{R}}
  \,.
\]

for the [[real vector space|real]] [[graded vector space]] of [[equivalence classes]] under the sign rules (eq:SignFromOrientationReversal) and (eq:SignRuleFromOrderingInternalVertices) in the [[linear span]] (eq:LinearSpanOfIsoClassesOfGraphs) of the [[isomorphism classes]] of graphs (Def. \ref{Graphs}) with $n$ external vertices.

For $\Gamma$ any graph, we write

\[
  \label{EquivalenceClassOfAGraphForSignRules}
  \big[ \Gamma \big]
  \;\in\;
  \widehat{Graphs}_n\big(\mathbb{R}^3\big)
  \,.
\]

for the [[equivalence class]] of the element in this [[real vector space|real]] [[graded vector space]] that it represents.

When there is not risk of confusion, we will still refer to this equivalence class $[\Gamma]$ as a _graph_.

=--

A further [[quotient space]] $Graphs_n(\mathbb{R}^3)$ of $\widehat Graphs_n(\mathbb{R}^3)$ (eq:VectorSpaceOfGraphsWithSignRulesImposed) will underly the actual graph complex [below](#TheGraphComplex).

+-- {: .num_example #GraphsWithDirectLoopsVanish}
###### Example
**(no tadpoles)**

A direct consequence of quotienting out the [[equivalence relation]]
(eq:SignFromOrientationReversal)
is that graphs with "[[tadpoles]]", namely with edges
that have coinciding [[source]] and [[target]] [[vertex]],
vanish in $\widehat Graphs_n(\mathbb{R}^3)$ (eq:VectorSpaceOfGraphsWithSignRulesImposed):

$$
  e \in Edg_\Gamma
  \;\text{s.t.}\;
  s(e) = t(e)
  \phantom{AA}
  \Rightarrow
  \phantom{AA}
  \big[ \Gamma \big]
  \;=\;
  0
$$

(This holds more generally for all $\mathbb{R}^D$ with odd $D  = 2k + 1$.)

=--

+-- {: .num_example #TrivalenGraphs}
###### Example
**(trivalent graphs)**

A graph without external vertices has degree (Def. \ref{DegreeOfAGraph}) equal to zero precisely if all its vertices (which are all internal, by assumption) are trivalent:

$$
  \#\!Verts_{ext} = 0
  \;\;\Rightarrow\;\;
  \big(
     deg(\Gamma) = 0
     \;\;\Leftrightarrow\;\;
     \text{all vertices are trivalent}
  \big)
  \,.
$$

The following is the smallest example of such a trivalent graph:

Hence in general, the degree of a trivalent graph is three times its number of external vertices:

$$
  \Gamma
  \;
  \text{trivalent}
  \;\;\;\Rightarrow \;\;\;
  deg\big( [\Gamma] \big)
  \;=\;
  3 \cdot \#\!Verts_{ext}
  \,.
$$

(Because such a graph is the result of taking a trivalent graph in degree 0 and replacing internal vertices of degree -3 with external vertices of degree 0.)

For example, the following graph, obtained from the above trivalent graph by replacing the three internal vertices labeled 1,2 and 3 with external vertices, is of degree $3 \cdot 3 = 9$:

=--

\linebreak

##### The algebra of graphs

+-- {: .num_defn #WedgeProductOfGraphs}
###### Definition
**(wedge product of graphs)**

For

$$
  [\Gamma_1], [\Gamma_2]
  \;\in\;
  \widehat Graphs_n\big( \mathbb{R}^D\big)
$$

two equivalence classes (eq:VectorSpaceOfGraphsWithSignRulesImposed) of isomorphism classes of graphs (Def. \ref{GradedVectorSpaceOfGraphs}) with the same number of external, their _wedge product_ is the element

$$
  [\Gamma_1] \wedge [\Gamma_2]
  \coloneqq
  [\Gamma_1 \wedge \Gamma_2]
  \;\in\;
  \widehat Graphs_n\big( \mathbb{R}^D\big)
$$

represented by the graph (Def. \ref{Graphs}) which is given by the [[disjoint union]] of [[linear orders]] of edges and _internal_ vertices

$$
  \Gamma_1 \wedge \Gamma_2
  \;\coloneqq\;
  \Big(
    Edgs_{\Gamma_1}
    \sqcup_{{}_{\lt}}
    Edgs_{\Gamma_2}
    \underoverset{t_1 \sqcup_{\lt} t_2}{s_1 \sqcup_{\lt} s_2}{\rightrightarrows}
    Vert_{ext}
    \sqcup_{{}_{\lt}}
    \big(
      Vert_{int,1}
      \sqcup_{{}_{\lt}}
      Vert_{int,2}
    \big)
  \Big)
  \,.
$$

=--

([Lambrechts-Volić 14, 6.3](#LambrechtsVolic14))

It is immediate that:

+-- {: .num_defn #GradedCommutativeAlgebraOfGraphs}
###### Lemma
**(graded commutative algebra of graphs)**

For $n \in \mathbb{N}$, the [[graded vector space]] $\widehat Graphs_n\big( \mathbb{R}^3\big)$ (eq:VectorSpaceOfGraphsWithSignRulesImposed) of graphs  (Def. \ref{GradedVectorSpaceOfGraphs}) equipped with the wedge product of graphs from Def. \ref{WedgeProductOfGraphs}

$$
  \array{
    \widehat{Graphs}_n(\mathbb{R}^3)
    \otimes
    \widehat{Graphs}_n(\mathbb{R}^3)
    &\overset{\wedge}{\longrightarrow}&
    \widehat{Graphs}_n(\mathbb{R}^3)
    \\
    \big(
      [\Gamma_1], [\Gamma_2]
    \big)
    &\mapsto&
    [\Gamma_1 \wedge \Gamma_2]
  }
$$

is a [[graded-commutative algebra]].

=--

\linebreak

##### The differential on graphs
 {#TheDifferential}

The [[differential]] on the graded vector space of equivalence classes of graphs (eq:VectorSpaceOfGraphsWithSignRulesImposed) is defined in terms of contraction of certain edges in the graph (Def. \ref{ContractionOfAnEdge} below). For this we first say which edges count as _contractible_ (Def. \ref{ContractibleEdges}) and which sign is picked up when contracting them (Def. \ref{SignOfAContractibleEdge}):

+-- {: .num_defn #ContractibleEdges}
###### Definition
**(contractible edges)**

Given a graph  $\Gamma$ (Def. \ref{Graphs}) we say that an [[edge]] $e \in Edg_\Gamma$ is a _contractible edge_ if the following conditions hold:

1. $e$ is not a [[tadpole]] (Example \ref{GraphsWithDirectLoopsVanish}):

   $$
     s(e) \neq t(e)
   $$

1. $e$ has at least one internal vertex:

   $$
     s(e) \in Vert_{int}
     \phantom{AA}
     \text{or}
     \phantom{AA}
     t(e) \in Vert_{int}
   $$

1. every internal vertex of $e$ is connected to more than just one other vertex, i.e. $e$ is _not_ a solid arrow in a diagram of the following form

We write

\[
  \label{SetOfContractibleEdges}
  Edg^{contr}_\Gamma
  \;\subset\;
  Egd_\Gamma
\]

for the [[subset]] of the contractible edges inside all edges of a graph $\Gamma$.

=--

([Lambrechts-Volić 14, p. 60](#LambrechtsVolic14))

+-- {: .num_defn #ContractionOfAnEdge}
###### Definition
**(contraction of an edge)**

Given a graph $\Gamma$ (Def. \ref{Graphs}) and a contractible edge $e \in Edg_\Gamma$ (Def. \ref{ContractibleEdges}) consider the induced graph obtained by removing that edge and the one of its vertices with the larger label

$$
  \Gamma / e
  \;\coloneqq\;
  \big(
    Edg_\Gamma
    \setminus \{e\}
  \big)
  \underoverset{q \circ t}{q \circ s}{\rightrightarrows}
  \big(
    Vert_\Gamma
    \setminus \{ max(s(e),t(e))  \}
  \big)
$$

and connecting all edges previously attached with the removed vertex to the remaining vertex of that edge (ie. the one with the smaller label):

$$
  \array{
    Vert_\Gamma &\overset{}{\longrightarrow}& Vert_{\Gamma/e}
    \\
    v &\mapsto&
   \left\{
    \array{
       min(s(e),t(e)) &\vert& \text{if}\; v = max(s(e),t(e))
       \\
       v &\vert& \text{otherwise}
    }
   \right.
  }
$$

=--

([Lambrechts-Volić 14, Def. 6.9](#LambrechtsVolic14))

+-- {: .num_defn #SignOfAContractibleEdge}
###### Remark
**(sign of a contractible edge)**

Given a _contractible edge_ $e \in Egd_\Gamma$ (Def. \ref{ContractibleEdges}) in some graph $\Gamma$ (Def. \ref{Graphs}) we associate a sign as follows:

$$
  \epsilon(e)
  \;\coloneqq\;
  \left\{
  \array{
    (-1)^{ t(e) } &\vert& t(e) \gt s(e)
    \\
    -(-1)^{s(e)} &\vert& s(e) \gt t(e)
  }
  \right.
  \,,
$$

where on the right we are identifying the [[linear order]] on the vertices with their [[natural number]]-labels as in (eq:NaturalNumberLabelsOfVertices).

=--

([Lambrechts-Volić 14, p. 65](#LambrechtsVolic14))

+-- {: .num_defn #DifferentialOnGraphs}
###### Definition
**(differential on graphs)**

For $Vert_{ext}$ a fixed [[finite set|finite]] [[linear order]] of $n = \#\! Vert_{ext}$ external vertices, define on the [[graded vector space]]
$\widehat Graphs_n\big( \mathbb{R}^3\big)$ (eq:VectorSpaceOfGraphsWithSignRulesImposed) a [[linear map|linear]] [[endomorphism]] $d$ given on the class $[\Gamma]$ \eqref{EquivalenceClassOfAGraphForSignRules} of a graph $\Gamma$ (Def. \ref{Graphs}) by the [[linear combination]] of all its edge contractions $\Gamma/e$ (Def. \ref{ContractionOfAnEdge}) for all contractible edges (Def. \ref{ContractibleEdges}) weighted by their sign (Def. \ref{SignOfAContractibleEdge}):

$$
  \array{
    \widehat Graphs_n\big( \mathbb{R}^3\big)
    &\overset{d}{\longrightarrow}&
    \widehat Graphs_n\big( \mathbb{R}^3\big)
    \\
    [\Gamma]
    &\mapsto&
    \underset{ e \in Edg^{contr}_\Gamma }{\sum}
    \epsilon(\Gamma) \cdot [\Gamma/e]
  }
  \,.
$$

=--

([Lambrechts-Volić 14, (84) ](#LambrechtsVolic14))

+-- {: .num_example #DifferentialOnLinearGraph}
###### Example
**(differential of propagator graph vanishes)**

The differential from Def. \ref{DifferentialOnGraphs}
vanishes on all graphs without any internal vertices (since these have no contractible edges in the sense of Def. \ref{ContractibleEdges}).
In particular it vanishes on the graph for the single [[Chern-Simons propagator]]:

=--

+-- {: .num_example #DifferentialOnLinearGraph}
###### Example
**(differential on linear graphs)**

The differential of Def. \ref{DifferentialOnGraphs} applied to the linear graph with one internal vertex:

Here the right hand side vanishes, due to the sign rule (eq:SignFromOrientationReversal).

While this example illustrates the general action of the differential, beware that this particular differential relation will not actually contribute to the graph complex, as the graph on the left is a "vanishing graph" in the sense of Def. \ref{VanishingGraphs} below (since the valence of the internal vertex is $\lt 3$).

=--

+-- {: .num_example #ThreeTermRelation}
###### Example
**(differential of trivalent diagram with single internal vertex)**

The image of the single trivalent internal vertex under the differential from Def. \ref{DifferentialOnGraphs} is as shown in the following:

Under the [[quasi-isomorphism]] (eq:TheQuasiIsomorphism) from the [[graph complex]] to the [[de Rham complex]] on the [[Fulton-MacPherson compactification]] of a [[configuration space of points]], given by sending each [[graph]] to its [[Chern-Simons propagator|Chern-Simons]] [[Feynman amplitude on compactified configuration spaces of points]] ([this Prop.](Fulton-MacPherson+operad#QuasiIsomorphismBetweenGraphComplexAndDeRhamComplexOfFM)), this relation becomes the "3-term relation" ([this Prop.](Fulton-MacPherson+operad#DeRhamCohomologyOfFMCompactification)):

$$
     \left[g_{i j}\right] \wedge \left[ g_{j k} \right]
     +
     \left[g_{j k}\right] \wedge \left[ g_{k i} \right]
     +
     \left[g_{k i}\right] \wedge \left[ g_{i j} \right]
     \;\sim\;
     0
$$

satisfied by the [[Chern-Simons propagator]] form

$$
  \left(
    g_{i j}
    \;\in\;
    \Omega^2_{PA}
    \Big(
      \underset{{}^{1,\cdots,n}}{Conf}
      \big(
        \mathbb{R}^D
      \big)
    \Big)
  \right)
  \,.
$$

=--

([Lambrechts-Volić 14, Figure 1 and 2](#LambrechtsVolic14))

+-- {: .num_example #DifferentialOfMinimalTrivalenVacuumGraph}
###### Example
**(Differential of minimal trivalent [[vacuum diagram]])**

The differential of the minimum trivalent [[vacuum diagram]] from Example \ref{TrivalenGraphs}:

=--

+-- {: .num_example #DifferentialOfTrivalentTreeWithTwoInternalVertices}
###### Example
**(differential of trivalent tree with two internal vertices)**

The differential (Def. \ref{DifferentialOnGraphs}) on the trivalent tree
with two internal vertices:

=--

+-- {: .num_lemma #DifferentialProperties}
###### Lemma
**(differential graded algebra structure)**

The [[linear map]] $d$ in Def. \ref{DifferentialOnGraphs} makes the [[graded vector space]] $\widehat Graphs_n(\mathbb{R}^D)$ a [[cochain complex]], and, with its [[graded algebra]]-[[structure]] from Lemma \ref{GradedCommutativeAlgebraOfGraphs} in fact a [[differential graded algebra]], in that:

1. (well defined) $d$ indeed only depends on the equivalence class $[\Gamma]$;

1. (degree) $d$ increases degree by +1

1. (nilpotency)

   $$
     d \circ d = 0
   $$

1. ([[derivation]])

   $$
     d( [\Gamma_1] \wedge [\Gamma_2] )
     \;=\;
     \big( d[\Gamma_1]  \big) \wedge [\Gamma_2]
     +
     (-1)^{deg(\Gamma_1)} [\Gamma_1] \wedge \big( d [\Gamma_2]\big)
     \,.
   $$

=--

([Lambrechts-Volić 14, Lemmas 6.11 - 6.14](#LambrechtsVolic14))

\linebreak

##### The graph complex
 {#TheGraphComplex}

+-- {: .num_defn #VanishingGraphs}
###### Definition
**(vanishing graphs)**

We say that a graph $\Gamma$ (Def. \ref{Graphs}) is a _vanishing graph_ if it is one or more of the following:

* it is a [[tadpole-diagram]]: $\Gamma$ contains an edge $e$ with $s(e) = t(e)$ (Example \ref{GraphsWithDirectLoopsVanish});

* it is a [[vacuum diagram]]: $\Gamma$ contains an internal vertex which is not connected, via some path of edges, to an external vertex;

* it has [[parallel edges]]: $\Gamma$ contains a pair $(a,b)$ of vertices, with more than one edge between them, i.e. if the [[preimage]] $(s,t)^{-1}\big\{(a,b)\big\}$ contains more than one element.

* it is less than trivalent: $\Gamma$ contains an internal vertex $v \in Verts_{int}$ that is less than trivalent, hence an internal vertex with less than 3 edges attached to it;

Write

$$
  {\widehat Graphs}^{nil}_n(\mathbb{R}^3)
  \;\subset\;
  \widehat Graphs_n\big( \mathbb{R}^3 \big)
$$

for the [[subspace]] spanned by vanishing graphs $[\Gamma]$ inside the [[graded vector space]] (eq:VectorSpaceOfGraphsWithSignRulesImposed) of all graphs.

=--

([Lambrechts-Volić 14, Def. 6.16](#LambrechtsVolic14))

+-- {: .num_lemma #VanishingGraphsAreDifferentialIdeal}
###### Lemma
**(vanishing graphs form a differential ideal)**

The subspace of vanishing graphs (Def. \ref{VanishingGraphs}) is a [[differential ideal]] in the [[differential graded-commutative algebra]] of all graphs (Lemma \ref{DifferentialProperties}).

=--

([Lambrechts-Volić 14, Lemma 6.17](#LambrechtsVolic14))

This implises that the quotient of all graphs by the vanishing graphs is still a differential graded-commutative algebra:

+-- {: .num_defn #GraphComplexDgcAlgebra}
###### Definition
**(graph complex)**

For $n \in \mathbb{N}$, the _graph complex_ on $n$ external vertices is the [[differential graded-commutative algebra]]

\[
  \label{TheGraphComplexAsQuotientByVanishingGraphs}
  Graphs_n
  \big(
    \mathbb{R}^3
  \big)
  \;\coloneqq\;
  \frac{
    {\widehat Graphs}_n\big( \mathbb{R}^3\big)
  }
  {
    {\widehat Graphs}^{nil}_n\big( \mathbb{R}^3\big)
  }
\]

of the [[differential graded-commutative algebra]] of all graphs (Lemma \ref{DifferentialProperties}) by its [[differential ideal]] (Lemma \ref{VanishingGraphsAreDifferentialIdeal}) of vanishing graphs (Def. \ref{VanishingGraphs}).

=--

([Lambrechts-Volić 14, Def. 6.19](#LambrechtsVolic14))

+-- {: .num_example #A9CocycleInGraphs3}
###### Example
**(A 9-cocycle in $Graphs_3(\mathbb{R}^3)$)**

The trivalent graph of degree 9 from Example \ref{TrivalenGraphs} is a [[cocycle]] in $Graphs_3(\mathbb{R}^3)$ (Def. \ref{GraphComplexDgcAlgebra}):

The computation of the differential is just as  for the 3-term relation in Example \ref{ThreeTermRelation},
but now all summands on the right have [[parallel edges]] and hence are vanishing graphs (Def. \ref{VanishingGraphs}) that are zero in the quotient (eq:TheGraphComplexAsQuotientByVanishingGraphs) defining the  graph complex $Graphs_3\big( \mathbb{R}^3 \big)$ (Def. \ref{GraphComplexDgcAlgebra}).

> This trivalent diagram must be exact, as there is not supposed to be any cohomology in degree 9 (by Prop. \ref{RealCohomologyOfConfigurationSpaceOfOrderedPointsInEuclideanSpace}). What's a trivializing coboundary?

=--

+-- {: .num_lemma #GraphsAreInNonNegativeDegree}
###### Lemma
**(graphs are in non-negative degree)**

For all $n \in \mathbb{N}$, the degrees (Def. \ref{DegreeOfAGraph})  of any non-vanishing graph $[\Gamma]$ in the graph complex $Graphs_n(\mathbb{R}^3)$ (Def. \ref{GraphComplexDgcAlgebra}) are non-[[negative number|negative]]

$$
  deg\big( [\Gamma] \big)
  \;\geq\;
  0
  \;\;
  \in
  \;\;
  Graphs_n(\mathbb{R}^3)
  \,.
$$

Moreover, the only graphs of degree 0 are those containg the external vertices and nothing else:

$$
  deg\big( [\Gamma]\big)
  =
  0
  \;\;\;\;\;\;\Leftrightarrow\;\;\;\;\;\;
  [\Gamma] = 0
  \;\;\;\text{or}\;\;\;
  [\Gamma] = \big[ {}_1 \;\; {}_2 \;\; \cdots \;\; {}_n  \big]
  \;\;\;\;\in\;\;
  Graphs_n(\mathbb{R}^3)
  \,.
$$

=--

+-- {: .proof}
###### Proof

Since [[tadpole]] graphs are vanishing graphs by Def. \ref{VanishingGraphs}, we may assume that each edge in $\Gamma$ has two distint vertices it ends on, for otherwise we have the zero element in the graph complex.

Now, since each edge has degree 2 (according to Def. \ref{DegreeOfAGraph}), we may think of each edge as contribution a degree +1 for each of its two vertices, via the coresponding half-edge ending on that vertex.

Since external vertices have degree 0, together with the half-edges that end on them they contribute a non-negative number to the total degree.

While internal vertices have degree -3 (according to Def. \ref{DegreeOfAGraph}), we have a vanishing graph if less than three edges meet at an internal vertex (by Def. \ref{VanishingGraphs}) and hence, once vanishing graphs have been quotiented out, also internal vertices together with the half-edges ending on them contribute a non-negative number to the  total degree.

This proves the first claim.

For the second claim, recall from Example \ref{TrivalenGraphs} that a graph has degree 0 precisely if it is a disjoint union of a) isolated external vertices and b) trivalent graphs all whose vertices are internal.

But if there is any disjoint non-empty sub-graph with only internal vertices, the total graph is again a vanishing graph by Def. \ref{VanishingGraphs}, since it contains internal vertices not connected to any external vertex.

Hence the only non-vanishing graphs of degree 0 are those which have some isolated external vertices and nothing else.

=--

\linebreak

#### $Graphs_n\big(  \mathbb{R}^2 \big)$

(...)

\linebreak

### **2)** Graph complexes modelling spaces of knots
  {#GraphComplexesModellingSpacesOfKnots}

We discuss graph complexes $KnotGraphs(\Sigma)$ modelling the [[real cohomology]] of [[spaces of knots]] $Emb(S^1,\Sigma)$.

For the case of $\Sigma = \mathbb{R}^D$ these were originally hinted at in [Kontsevich 93, Section 5](#Kontsevich93) and discussed in detail in [Cattaneo, Cotta-Ramusino, Longoni 02](#CattaneoCottaRamusinoLongoni02) [05](#CattaneoCottaRamusinoLongoni05), see also [Bar-Natan 91](#BarNatan91). Review is in [Volić 13, Section 4](#Volic13).

\linebreak

#### $KnotGraphs\big( \mathbb{R}^3 \big)$
 {#KnotGraphsDefinition}

We state the definition of the graph complexes $KnotGraphs\big( \mathbb{R}^3\big)$ of _knot graphs_ (according to the [above](#IdeaGraphComplexModelForSpacesOfKnots)) associated with [[Euclidean space]] of [[dimension]] 3, following [Cattaneo, Cotta-Ramusino, Longoni 02](#CattaneoCottaRamusinoLongoni02) [05](#CattaneoCottaRamusinoLongoni05).

The definition is almost exactly the same as that of $\oplus_n Graphs_n\big( \mathbb{R}^3\big)$ [above](#GraphComplexForConfigurationsOfPointsInR3), except for the following two differences:

**1)** external vertices now carry degree -1, so that the formula (eq:DegreeOfAGraph) for the degree of a graph is changed to

\[
  \label{DegreeOfAKnoptGraph}
  deg\big( [\Gamma]\big)
  \;=\;
  (D-1) \cdot \#\!Edges
  \;
  - D \cdot \#\!Vertices_{int}
  \;
  - 1 \cdot \#\!Vertices_{ext}
  \,,
\]

([CCRL 02, (4.6)](#CattaneoCottaRamusinoLongoni02), [Volić 13, Def. 4.3](#Volic13))

**2)** There is implicit a contractible edge from the $k$th to the $(k+1)$st external vertex (an _arc_), in that the definition of the differential (Def. \ref{DifferentialOnGraphs}) regards these as contractible edges.

([CCRL 02, 4.2](#CattaneoCottaRamusinoLongoni02), [Volić 13, p. 35](#Volic13))

**3)** Apparently it must be understood that graphs whose labelling of external vertices differs by a cyclic permutation are identified. (?!)

(...)

+-- {: .num_example #KnotGraphCocycleOfOrder2}
###### Example
**(knot graph cocycle of order 2)**

The differential in $KnotGraphs(\mathbb{R}^3)$ of the single trivalent internal vertext is a variant of that in $Graphs_3\big(\mathbb{R}^3\big)$ (Example \ref{ThreeTermRelation}) and that in $Graphs_0\big( \mathbb{R}^3\big)$ (Example \ref{DifferentialOfMinimalTrivalenVacuumGraph}):

Notice how the edges parallel to the outer dashed arcs do not make the graphs in the first line vanish, but the [[parallel edges]] appearing in the second line make vanishing graphs (Def. \ref{VanishingGraphs}).

In contrast, the differential of the diagram in $KnotGraphs(\mathbb{R}^3)$ consisting of just two overlapping chords gets contributions only from contraction of the four arcs:

In both cases, the identification of graphs whose external labels are ccyclically permutated identifies the right hand sides of both these coboundaries with multiples of one and the same element in the graph complex.

Accordingly, the corresponding linear combination

is a [[cocycle]] of degree 0 in $KnotGraphs(\mathbb{R}^3)$:

=--

([Bar-Natan 91, 4.3.2](#BarNatan91), [Cattaneo, Cotta-Ramusino, Longoni 02, Figure 2](#CattaneoCottaRamusinoLongoni02))

\linebreak
\linebreak

$\in H^0\Big( KnotGraphs\big( \mathbb{R}^3 \big) \Big)$

For the corresponding [[Feynman amplitude]] [[knot invariant]] see Prop. \ref{FirstKnotInvariantFromChernSimonsFeynmanAmplitude} below.

\linebreak

## Properties
 {#Properties}

> under construction

### Cohomology of configuration spaces via Feynman diagrams
 {#RelationToConfigurationSpaces}

Reading the graphs as [[Chern-Simons theory]] [[Feynman diagrams]], hence as instructions for [[configuration space of points|configuration space]]-[[integrals]] of [[wedge products]] of [[Feynman propagators]] assigned to [[edges]] in a graph, with [[propagators regarded as differential forms on configuration spaces]],
yields a [[quasi-isomorphism]] from the graph complex to the [[real cohomology]] of [[configuration spaces of points]].

([Lambrechts-Volić 14, Section 9](#LambrechtsVolic14))

\[
  \underset{
    \color{blue}
    \array{
      \phantom{A}
      \\
      \text{graph complex}
      \\
      \text{of n-point Feynman diagrams}
      \\
      \text{for Chern-Simons theory}
      \\
      \text{on} \; \Sigma
    }
  }{
    Graphs_n(\Sigma)
  }
  \underoverset{
    \simeq_{\mathrlap{qi}}
  }
  {
    \color{blue}
    \array{
      \text{assign Feynman amplitudes}
      \\
      \text{of Chern-Simons theory}
      \\
      \text{by configuration-space integral}
      \\
      \text{of internal vertices over}\; \Sigma
      \\
      \phantom{A}
    }
  }
  {
    \longrightarrow
  }
  \underset{
    \color{blue}
    \array{
      \phantom{A}
      \\
      \text{de Rham algebra}
      \\
      \text{of semi-algebraic differential forms}
      \\
      \text{on the FM-compactification}
      \\
      \text{of the configuration space of n points}
      \\
      \text{in}\; \Sigma
    }
  }{
  \Omega^\bullet_{PA}
  \big(
    \underset{{}^{1,\cdots,n}}{Conf}\big(  \Sigma \big)
  \big)
  }
  \,.
\]

+-- {: .num_prop #RealCohomologyOfConfigurationSpaceOfOrderedPointsInEuclideanSpace}
###### Proposition
**([[real cohomology]] of configuration spaces of ordered points in [[Euclidean space]])**

The [[real cohomology|real]] [[cohomology ring]] of the configuration spaces

$$
  \underset{{}^{1,\cdots,n}}{Conf}\big( \mathbb{R}^D \big)
  \;\coloneqq\;
  \big( \mathbb{R}^D \big)^n \setminus FatDiag
$$

of $n$ ordered points in [[Euclidean space]] $\mathbb{R}^D$ is [[generators and relations|generated]] by elements

$$
  \omega_{i j}
  \;\;
  \in
  H^2
  \Big(
    \underset{{}^{1,\cdots,n}}{Conf}\big( \mathbb{R}^D \big),
    \mathbb{R}
  \Big)
$$

for $i, j \in \{1, \cdots, n\}$

subject to these three [[generators and relations|relations]]:

1. $\omega_{i j} = (-1)^D \omega_{j i} $;

1. $\omega_{i j} \wedge \omega_{i j} \;=\; 0$;

1. $\omega_{i j} \wedge \omega_{j k} + \omega_{j k} \omega_{k i} + \omega_{k i} \wedge \omega_{i j} = 0$.

Hence:

$$
  H^\bullet
  \Big(
    \underset{{}^{1,\cdots,n}}{Conf}\big( \mathbb{R}^D \big),
    \mathbb{R}
  \Big)
  \;\simeq\;
  \mathbb{R}\Big[ \big\{\omega_{i j} \big\}_{i, j \in \{1, \cdots, n\}} \Big]
  \Big/
  \left(
    \array{
      \omega_{i j} = (-1)^D \omega_{j i}
      \\
      \omega_{i j} \wedge \omega_{i j} = 0
      \\
      \omega_{i j} \wedge \omega_{j k} + \omega_{j k} \omega_{k i} + \omega_{k i} \wedge \omega_{i j} = 0
     }
     \;\;
     \text{for}\;
     i,j \in \{1, \cdots, n\}
  \right)
$$

=--

This is due to [Arnold 69](configuration+space+of+points#Arnold69), [Cohen 73](configuration+space+of+points#Cohen73).

| [[real cohomology]] of [[configuration space of points]] |  [[graph complex]] ([this Prop.](configuration+space+of+points#RealCohomologyOfConfigurationSpaceOfOrderedPointsInEuclideanSpace)) |
|------------------|-------------------------|
| generator
 $\omega_{i j} \in H^2\Big( \underset{{}^{1,\cdots,n}}{Conf}\big( \mathbb{R}^3\big) \Big)$ | [[edge]]
 $i \longrightarrow j$   |
| **[[generators and relations|relations]]:** | **[[generators and relations|relations]]:** |
| $\omega_{i j} = - \omega_{j i} $ | graph changes sign when edge is reversed (Def. \ref{SignRulesForGraphs}) |
| $\omega_{i j} \wedge \omega_{i j} \;=\; 0 $ | graph with [[parallel edges]] vanishes (Def. \ref{VanishingGraphs}) |
| $\omega_{i j} \wedge \omega_{j k} + \omega_{j k} \omega_{k i} + \omega_{k i} \wedge \omega_{i j} = 0$ | coboundary of trivalent vertex (Example \ref{ThreeTermRelation}) |

\linebreak

### Cohomology of knot space / higher Vassiliev invariants

The [[Feynman amplitudes]] of knot graphs (as [above](#KnotGraphsDefinition)) are  [[Vassiliev knot invariants]].

This was originally hinted at in [Kontsevich 93, Section 5](#Kontsevich93).
Details are in [Cattaneo, Cotta-Ramusino, Longoni 02](#CattaneoCottaRamusinoLongoni02).
Review is in [Volić 13, Section 4](#Volic13).

\[
  \underset{
    \color{blue}
    \array{
      \phantom{A}
      \\
      \text{graph complex}
      \\
      \text{of n-point Feynman diagrams}
      \\
      \text{for Chern-Simons theory}
      \\
      \text{on} \; \Sigma
    }
  }{
    KnotGraphs(\Sigma)
  }
  \underoverset{
  }
  {
    \color{blue}
    \array{
      \text{assign Feynman amplitudes}
      \\
      \text{of Chern-Simons theory}
      \\
      \text{by configuration-space integral}
      \\
      \text{of internal vertices over}\; \Sigma
      \\
      \text{and of external vertices along the knot}
      \\
      \phantom{A}
    }
  }
  {
    \longrightarrow
  }
  \underset{
    \color{blue}
    \array{
      \phantom{A}
      \\
      \text{de Rham algebra of}
      \\
      \text{space of knots in}\; \Sigma
    }
  }{
  \Omega^\bullet_{PA}
  \big(
    Emb\big( S^1, \Sigma\big)
  \big)
  }
  \,.
\]

+-- {: .num_prop #FirstKnotInvariantFromChernSimonsFeynmanAmplitude}
###### Proposition
**([[knot invariant]] from [[Chern-Simons theory]] [[Feynman amplitude]])**

The [[Feynman amplitude]]
of the knot graph cocycle [[Feynman diagram]] from Example \ref{KnotGraphCocycleOfOrder2}, in [[Chern-Simons theory]] with a [[Wilson loop]] [[knot]], is a [[knot invariant]]:

=--

This is due to [Guadagnini-Martellini-Mintchev 89](#GuadagniniMartelliniMintchev89)
and [Bar-Natan91](#BarNatan91), recalled im [Volić 13, Theorem 3.3.5](#Volic13).

\linebreak

Various authors discuss [[Vassiliev invariants]] in terms of graphs subject to the "[[STU-relation]]"

([Kontsevich 93, Figure 8](#Kontsevich93), [Bar-Natan 95, Figure 3](Vassiliev+invariant#BarNatan95)) and the "IHX-relation"  which it implies ([Bar-Natan 95, Theorem 6](Vassiliev+invariant#BarNatan95))

> graphics grabbed from [Kontsevich 93](#Kontsevich93)

That these relations characterize the cohomology of the knot-graph complex in the respective degrees is shown in [Koytcheff-Munson-Volic 13, Section 3.4](#KoytcheffMunsonVolic13)

> graphics grabbed from [Koytcheff-Munson-Volic 13](#KoytcheffMunsonVolic13)

### Cohomology of knot graph complex is weight systems on chord diagrams

[[cohomology of knot graph complex is weight systems on chord diagrams]]

## Further applications

...moduli spaces

...deformation theory

...Rozansky-Witten theory

...description of the classifying space $BOut(F_n)$ of the group of outer automorphisms of a free group with $n$ generators

... another graph complex controls the universal $L_\infty$-deformations of the space of polyvector fields.

## Related concepts

* [[configuration space of points]], [[space of knots]]

* [[Fulton-MacPherson operad]]

* [[Chern-Simons propagator]]

* [[correlator as differential form on configuration space of points]]

* [[Kontsevich formality]]

* [[Rozansky-Witten theory]]

* [[formal noncommutative symplectic geometry]]

## References

### General

The rough definition of the graph complex, and its relation to [[Chern-Simons theory]] [[Feynman amplitudes]] and [[Vassiliev knot invariants]] was sketched in

* {#Kontsevich92} [[Maxim Kontsevich]], pages 11-12 of _Feynman diagrams and low-dimensional topology_, First European Congress of Mathematics, 1992, Paris, vol. II, Progress in Mathematics __120__, Birkhäuser (1994), 97–121 ([pdf](http://www.ihes.fr/~maxim/TEXTS/Feynman%20%20diagrams%20and%20low-dimensional%20topology.pdf))

* {#Kontsevich93} [[Maxim Kontsevich]], Section 5 of: _Vassiliev's knot invariants_, Advances in Soviet Mathematics, Volume 16, Part 2, 1993 ([pdf](http://pagesperso.ihes.fr/~maxim/TEXTS/VassilievKnot.pdf), [[KontsevichVassilievInvariants93.pdf:file]])

* {#Kontsevich99b} [[Maxim Kontsevich]], around Def. 15 and Lemma 3 in _Operads and Motives in Deformation Quantization_, Lett. Math. Phys. 48 35-72, 1999 ([arXiv:math/9904055](https://arxiv.org/abs/math/9904055))

### Surveys

* Marko Berghoff, _Graph complexes from the geometric viewpoint_, [arXiv](https://arxiv.org/abs/2312.11237).

### As a rational model for configuration spaces

A clean account and proof of the graph complex as a model for the [[rational homotopy type]] of the [[Fulton-MacPherson compactification]] of [[configuration spaces of points]] (exhibiting the [[formal smooth manifold|formality]] of the [[little n-disk operads]]) is in

* {#LambrechtsVolic14} [[Pascal Lambrechts]], [[Ismar Volić]], sections 6 and 7 and 9 of: _Formality of the little N-disks operad_, Memoirs of the American Mathematical Society no. 1079, 2014  ([arxiv:0808.0457](https://arxiv.org/abs/0808.0457), [doi:10.1090/memo/1079](http://dx.doi.org/10.1090/memo/1079))

Discussion as a direct model for the rationalized [[homotopy groups]] of the [[configuration space of points]]:

* [[Pascal Lambrechts]], [[Victor Tourtchine]], _Homotopy graph-complex for configuration and knot spaces_, Transactions of the AMS, Volume 361, Number 1, January 2009, Pages 207–222 ([arxiv:math/0611766](https://arxiv.org/abs/math/0611766))

further review:

* {#Fresse18} [[Benoit Fresse]], _Little discs operads, graph complexes and Grothendieck--Teichmüller groups_, in [[Haynes Miller]] (ed.) _[[Handbook of Homotopy Theory]]_ ([arXiv:1811.12536](https://arxiv.org/abs/1811.12536))

Further discussion of the graph complex as a model for the [[de Rham cohomology]] of  [[configuration spaces of points]] is in

* [[Najib Idrissi]], _The Lambrechts-Stanley Model of Configuration Spaces_, Invent. Math, 2018 ([arXiv:1608.08054](https://arxiv.org/abs/1608.08054), [doi:10.1007/s00222-018-0842-9](https://dx.doi.org/10.1007/s00222-018-0842-9))

* {#CamposWillwacher16} [[Ricardo Campos]], [[Thomas Willwacher]], _A model for configuration spaces of points_ ([arXiv:1604.02043](https://arxiv.org/abs/1604.02043))

* {#Campos17} [[Ricardo Campos]], _Batalin-Vilkovisky formality and configuration spaces of points_, 2017 ([doi:10.3929/ethz-a-010886114](https://doi.org/10.3929/ethz-a-010886114))

* {#CamposIdrissiLambrechtsWillwacher18} [[Ricardo Campos]], [[Najib Idrissi]], [[Pascal Lambrechts]], [[Thomas Willwacher]], _Configuration Spaces of Manifolds with Boundary_ ([arXiv:1802.00716](https://arxiv.org/abs/1802.00716))

* [[Ricardo Campos]], Julien Ducoulombier, [[Najib Idrissi]], [[Thomas Willwacher]], _A model for framed configuration spaces of points_ ([arXiv:1807.08319](https://arxiv.org/abs/1807.08319))

See also

* {#Kontsevich99a} [[Maxim Kontsevich]], _Rozansky–Witten invariants via formal geometry_, Compositio Mathematica __115__: 115–127, 1999, [doi](http://dx.doi.org/10.1023/A:1000619911308), [arXiv:dg-ga/9704009](http://arxiv.org/abs/dg-ga/9704009)

* [[Andrey Lazarev]], _Operads and topological conformal field theories_, [pdf](http://www.maths.lancs.ac.uk/~lazarev/1-10.pdf); and older versio: _Graduate lectures on operads and topological field theories_, zip file with 11 pdfs, [over 5 Mb](http://www2.le.ac.uk/departments/mathematics/extranet/staff-material/staff-profiles/al179/New%20Folder.zip)

* [[Alastair Hamilton]], _A super-analogue of Kontsevich's theorem on graph homology_, Lett. Math. Phys. __76__ (2006), no. 1, 37--55, [math.QA/0510390](http://arxiv.org/abs/math/0510390)

* A. Lazarev, [[Alexander Voronov]], _Graph homology: Koszul and Verdier duality_ ([math.QA/0702313](http://arxiv.org/abs/math/0702313))

* [[Mikhail Movshev]], _A definition of graph homology and graph K-theory of algebras_ ([math.KT/9911111](http://arxiv.org/abs/math/9911111))

* [[Kiyoshi Igusa]], _Graph cohomology and Kontsevich cycles_, Topology __43__ (2004), n. 6, p. 1469-1510, [MR2005d:57028](http://www.ams.org/mathscinet-getitem?mr=2005d:57028), [doi](http://dx.doi.org/10.1016/j.top.2004.03.004)

* [[Vasily Dolgushev]], [[Christopher Rogers]], [[Thomas Willwacher]], _Kontsevich's graph complex, GRT, and the deformation complex of the sheaf of polyvector fields_ ([arxiv/1211.4230](http://arxiv.org/abs/1211.4230))

* [[Damien Calaque]], Carlo A. Rossi, _Lectures on [[Duflo isomorphism]]s in Lie algebra and complex geometry_, European Math. Soc. 2011

* [[Sergei Merkulov]], _Graph complexes with loops and wheels_, in (Manin's Festschrift:) Algebra, Arithmetic, and Geometry, Progress in Mathematics __270__ (2009) 311-354, [doi](http://dx,doi.org/10.1007/978-0-8176-4747-6_10), [pdf](http://www2.math.su.se/~sm/Papers/graph_complexes.pdf)

* [[Martin Markl]], [[Sergei Merkulov]], S. Shadrin, _Wheeled PROPs, graph complexes and the master equation_, J. Pure Appl. Algebra, 213(4):496–535, 2009, [math.AT/0610683](https://arxiv.org/abs/math/0610683)

The following survey has discussion of context between the graph complex and [[Batalin-Vilkovisky formalism]]:

* Jian Qiu, Maxim Zabzine, _Introduction to graded geometry, Batalin-Vilkovisky formalism and their applications_, [arxiv/1105.2680](http://arxiv.org/abs/1105.2680)
* Jian Qiu, Maxim Zabzine, _Knot weight systems from graded symplectic geometry_, [arxiv/1110.5234](http://arxiv.org/abs/1110.5234)

* [[Alastair Hamilton]], [[Andrey Lazarev]], _Graph cohomology classes in the [[BV-BRST formalism|Batalin-Vilkovisky formalism]]_, J.Geom.Phys. __59__:555-575, 2009, [arxiv/0701825](http://arxiv.org/abs/math/0701825)

### As construction of higher order Vassiliev invariants

Discussion of the graph complex as computing higher order [[Vassiliev invariants]], hence the [[real cohomology]] of spaces of [[knots]] (with the cohomology in degree-0 being the ordinary [[Vassiliev invariants]]):

* [Kontsevich 93, Section 5](#Kontsevich93)

* Daniel Altschuler, Laurent Freidel, _Vassiliev knot invariants and Chern-Simons perturbation theory to all orders_, Commun. Math. Phys. 187 (1997) 261-287 ([arxiv:q-alg/9603010](https://arxiv.org/abs/q-alg/9603010))

* {#CattaneoCottaRamusinoLongoni02} [[Alberto Cattaneo]], Paolo Cotta-Ramusino, Riccardo Longoni, _Configuration spaces and Vassiliev classes in any dimension_, Algebr. Geom. Topol. 2 (2002) 949-1000 ([arXiv:math/9910139](https://arxiv.org/abs/math/9910139))

* {#CattaneoCottaRamusinoLongoni05} [[Alberto Cattaneo]], Paolo Cotta-Ramusino, Riccardo Longoni, _Algebraic structures on graph cohomology_, Journal of Knot Theory and Its Ramifications, Vol. 14, No. 5 (2005) 627-640 ([arXiv:math/0307218](https://arxiv.org/abs/math/0307218), [doi:10.1142/S0218216505004019](http://dx.doi.org/10.1142/S0218216505004019), [math.GT/0307218](http://arxiv.org/abs/math/0307218), [MR2006g:58021](http://www.ams.org/mathscinet-getitem?mr=2006g:58021))

* {#GuadagniniMartelliniMintchev89} E. Guadagnini, E. Martellini, M. Mintchev, _Chern-Simons field theory and  link invariants_,  In:   _Knots,  Topology  and  QuantumField Theories_,  Proceedings of the Johns Hopkins Workshop onCurrent Problems in Particle Theory 13, Florence (1989), World Scientific ([doi:10.1142/9789814540742](https://doi.org/10.1142/9789814540742))

* {#BarNatan91} [[Dror Bar-Natan]], _Perturbative aspects of the Chern-Simons topological quantum field theory_, thesis 1991 ([spire:323500](http://inspirehep.net/record/323500), [proquest:303979053](https://search.proquest.com/docview/303979053), [[BarNatanPerturbativeCS91.pdf:file]])

Reviewed in:

* {#Volic13} [[Ismar Volić]], Section 4 of: _Configuration space integrals and the topology of knot and link spaces_, [Morfismos, Vol 17, no 2, 2013](https://fdocuments.co/amp/document/morfismos-vol-17-no-2-2013.html) ([arxiv:1310.7224](https://arxiv.org/abs/1310.7224))

### Coproduct

Discussion of [[coproducts]] on a graph complex, given by decomposition of graphs into subgraphs and contractions of subgraphs:

* {#Ionescu03} [[Lucian Ionescu]], _Perturbative Quantum Field Theory and Configuration Space Integrals_, In: John Byden, _Advances in Topological Quantum Field Theory_ ([arXiv:hep-th/0307062](https://arxiv.org/abs/hep-th/0307062), [doi:10.1007/978-1-4020-2772-7](https://link.springer.com/book/10.1007/978-1-4020-2772-7))

* {#Ionescu05} [[Lucian Ionescu]], _Cohomology of Feynman graphs and perturbative quantum field theory_, In: O. Kovras, _Focus on Quantum Field Theory_, Nova Publishers Inc. 2004,  ([arXiv:math/0506142](https://arxiv.org/abs/math/0506142))

### Cohomology
 {#ReferencesCohomology}

On the [[cochain cohomology]] of [[graph complexes]]:

Characterization of cohomology of (...) in terms of [[STU-relations]] and HKX-relations:

* {#KoytcheffMunsonVolic13} Robin Koytcheff, Brian A. Munson, [[Ismar Volić]], Section 3.4 of: _Configuration space integrals and the cohomology of the space of homotopy string links_, J. Knot Theory Ramif. 22, no. 11, 73 pp. (2013) ([arXiv:1109.0056](https://arxiv.org/abs/1109.0056))

Concrete examples of cohomology classes in certain bi-degrees in $Graphs_0(\mathbb{R}^2)$ by [[computer experiment]]:

* {#BarNatanMcKay} [[Dror Bar-Natan]], [[Brendan McKay]], _Graph cohomology -- An overview and some computations_ ([[BarNatanMcKayGraphCohomology.pdf:file]])

$H^0\big( Graphs_0(\mathbb{R}^2) \big)$ is the [[Lie algebra]] of the [[Grothendieck-Teichmüller group]]:

* {#Willwacher10} [[Thomas Willwacher]], _M. Kontsevich's graph complex and the Grothendieck-Teichmueller Lie algebra_, Invent. math. (2015) 200: 671 ([arxiv:1009.1654](http://arxiv.org/abs/1009.1654))

* {#DolgushevRogers12} [[Vasily Dolgushev]], [[Christopher Rogers]], _Notes on Algebraic Operads, Graph Complexes, and Willwacher's Construction_, In: Mathematical aspects of quantization 583 (2012): 25-145. ([arXiv:1202.2937](https://arxiv.org/abs/1202.2937))

* [[Vasily Dolgushev]], _A manifestation of the Grothendieck-Teichmuellergroup in geometry_ ([slides pdf](https://math.temple.edu/~vald/AMSslides.pdf))

See also:

* Kevin Morand, _A note on multi-oriented graph complexes and deformation quantization of Lie bialgebroids_ ([arXiv:2102.07593](https://arxiv.org/abs/2102.07593))

More:

* {#KoroshkinWillwacherZivkovic14} Anton Khoroshkin, [[Thomas Willwacher]], Marko Živković, _Differentials on graph complexes_ ([arXiv:1411.2369](https://arxiv.org/abs/1411.2369))

### More

In relation to [[Feynman rules]]:

* Marko Berghoff, [[Dirk Kreimer]], _Graph complexes and Feynman rules_ ([arXiv:2008.09540](https://arxiv.org/abs/2008.09540))

[[!redirects graph complexes]]

[[!redirects graph homology]]
[[!redirects graph homology group]]
[[!redirects graph homology groups]]

[[!redirects graph cohomology]]
[[!redirects graph cohomology group]]
[[!redirects graph cohomology groups]]

[[!redirects knot graph complex]]
[[!redirects knot graph complexes]]
[[!redirects knot graph cohomology]]

[[!redirects knot-graph complex]]
[[!redirects knot-graph complexes]]


graph minor

#Contents#
* automatic table of contents goes here
{:toc}

## Idea

Let $G$ and $H$ be [[graph|simple graphs]]. Then $H$ is a **graph minor** of $G$ if it is isomorphic to a graph obtained by applying a sequence of operations of the following sort, starting from $G$:

* Removing edges;

* Removing isolated points;

* Contracting edges, where contracting an edge means removing it and identifying its endpoints.

Graph minors can be viewed as certain types of subquotients in an appropriate category of graphs; this article describes which subquotients those are in categorical language.

+-- {: .num_remark}
###### Disclaimer
To be honest, it's not really clear how a categorical POV (or at least the POV presented here) can usefully inform or guide or clarify graph theory. For the time being, we are just collecting some notes. No deep theorems of graph theory are proved here. The main observation is simply that graphs form a category with very good properties, and there may be potential of putting those properties to effective use some day.
=--

## Categorical POV

We take the point of view that a *simple graph* (consisting of a set of vertices $V$ and a collection of two-element subsets of $V$ called "edges") carries the same information as a set $V$ equipped with a reflexive symmetric relation $E$, and that a morphism of simple graphs should be defined as a function between the vertex-sets which respects such relations. Indeed, such a relation determines and is uniquely determined by a simple graph $G$ where for given vertices $x, y \in V$, there is an edge $\{x, y\}$ between $x$ and $y$ in $G$ iff both $(x, y) \in E$ and $x \neq y$. We will write $E(x, y)$ to mean $(x, y) \in E$.

+-- {: .num_remark}
###### Remark
Since edges are $2$-element subsets, it might seem more natural to interpret a simple graph as a set $V$ equipped with a symmetric *irreflexive* relation $E$, whereby $E(x, x)$ is false for all $x \in V$. But the result is a rather poor category which lacks certain morphisms we would like, such as edge contractions, at least if we take the morphisms $(V, E) \to (W, F)$ to be functions $f: V \to W$ which preserve those relations. Indeed, we could never have both $E(x, y)$ and $f(x) = f(y)$ since irreflexivity of $F$ would render $F(f(x), f(y))$ false. Thus, requiring reflexivity is the more flexible option.
=--

Notice that contraction of edges yields a quotient in this category. For example, if we want to contract a collection of edges by identifying certain points along an equivalence relation $R$, that is via some quotient map $q: V \to V/R$, then we simply take the image of the composite

$$E \hookrightarrow V \times V \stackrel{q \times q}{\to} V/R \times V/R;$$

notice this image is a reflexive symmetric relation on $V/R$.

Thus, we let $SimpGph$ denote the category of sets equipped with a reflexive symmetric relation. This category is convenient in many respects (for details see [[category of simple graphs]]):

+-- {: .num_theorem}
###### Theorem
The category $SimpGph$ has the following properties.

* It is a [[Grothendieck quasitopos]]. In particular, both it and its opposite $SimpGph^{op}$ are regular categories.

* It is an $\infty$-[[extensive category]].

* The forgetful functor $\Gamma = \hom(1, -): SimpGph \to Set$ is faithful and exhibits $SimpGph$ as a [[topological concrete category]].

* $SimpGph$ is *cohesive*: the adjoint string $\Delta \dashv \Gamma \dashv \nabla: Set \to SimpGph$ has a further left adjoint $\Pi \dashv \Delta$, and $\Pi: SimpGph \to Set$ preserves products.
=--

Of course $\Delta$ assigns to a set $X$ the discrete graph on $X$ (the minimal reflexive relation $\delta_X \hookrightarrow X \times X$), and $\nabla$ the codiscrete graph (the maximal relation $1_{X \times X}$), while $\Pi(G)$ is the set of connected components of $G$. Let $\gamma: 1 \to \Delta \Pi$ denote the unit of $\Pi \dashv \Delta$; the component $\gamma G: G \to \Delta \Pi (G)$ takes a vertex $v \in G$ to the connected component to which it belongs. Notice that $\gamma G$ is a regular epi.

+-- {: .num_defn}
###### Definition
A *contraction* map between simple graphs is a morphism $\pi: G \to G'$ that arises by pushing out a map of the form $\gamma K$ along some map $f: K \to G$:

$$\array{
K & \stackrel{f}{\to} & G \\
\mathllap{\gamma K} \downarrow & po & \downarrow \mathrlap{\pi} \\
\Pi(K) & \to & G'
}$$

is a pushout square. If $G \to G'$ is a contraction map, we also say $G'$ is a *contraction* of $G$.
=--

It is immediate that a contraction map is a regular epi, since every $\gamma K$ is regular epi.

+-- {: .num_prop #composition}
###### Proposition
Contraction maps are closed under composition.
=--

+-- {: .proof}
###### Proof

=--

+-- {: .num_defn}
###### Definition
A graph $H$ is a *minor* of a graph $G$ if it arises as a subobject of some contraction of $G$:

$$\array{
 & & G \\
 & & \downarrow \mathrlap{contraction} \\
H & \rightarrowtail & G'
}$$

=--

+-- {: .num_prop}
###### Proposition
The subquotient relation is reflexive and transitive.
=--

+-- {: .proof}
###### Proof
Reflexivity is clear. For transitivity, we compose subquotients by taking a pushout square as follows.

$$\array{
G & \twoheadrightarrow & G' & \twoheadrightarrow & G'' \\
 & &  ^\mathllap{mono} \uparrow & & \uparrow^\mathrlap{mono} \\
 & & H & \twoheadrightarrow & H' \\
 & & & & \uparrow^\mathrlap{mono} \\
 & & & & J
}$$

where we use the simple fact that the pushout of a mono along an arrow in $SimpGph$ is a mono (because $Vert$ reflects monos and preserves monos and pushouts, plus the fact that the pushout of a mono in $Set$ is a mono).
=--

For finite graphs, the subquotient relation is also antisymmetric (if $G$ and $H$ are minors of one another, then they are isomorphic). Indeed, if either the arrow $G \twoheadrightarrow G'$ or the arrow $H \hookrightarrow G'$ is not an isomorphism, then $H$ has strictly fewer edges and vertices than $G$. This is clearly not the case for infinite graphs (e.g., the infinite rooted binary tree without leaves contains as a subgraph a disjoint sum of two copies of itself).

(To be expanded, eventually. Hopefully one can develop a categorical story about graph minors in particular.)

## Minor-closed families

* The collection of [[forests]] is closed under the graph minor relation.

* The collection of [[planar graphs]] is closed under the graph minor relation.

## Graph minor theorem

This celebrated result of Robertson and Seymour says that no collection of finite graphs, ordered by the graph minor relation, has an infinite [[antichain]]. Equivalently, that in any collection of finite graphs, there are only finitely many elements which are minimal with respect to the graph minor relation.

As a corollary, any class $F$ of finite graphs which is closed under minors admits a **forbidden minor characterization**: is precisely the class of graphs which do not have any members of a specific finite family of graphs
as a minor. (Proof: the complement of $F$ in the poset of finite graphs has finitely many minimal members. Then, $x$ belongs to $F$ iff none of these minimal members occurs as a minor of $x$.

For example, the class of forests is precisely the class of finite graphs which do not have a triangle $K_3$ as a minor. The class of planar graphs is precisely the class of finite graphs which do not have $K_5$ or $K_{3, 3}$ as a minor.

Forbidden minor characterizations also exist for certain classes of [[matroid|matroids]]. (See for example Wikipedia [here](http://en.wikipedia.org/wiki/Matroid#Forbidden_minors).)

## References

* Wikipedia article on graph minors [(link)](http://en.wikipedia.org/wiki/Minor_%28graph_theory%29)

[[!redirects graph minors]]


graph morphism

Let $C$ be a [[category]] admitting finite [[fibred products]] and $f : X \to Y$ be a [[morphism]] over some base $S$.  The **graph morphism of $f$** is the [[morphism]]

$$
  \Gamma_f \colon X \to X \times_S Y
$$

induced by $\mathbf{1}_X : X \to X$ and $f : X \to Y$.

The [[image]] of the graph morphism of $f$ is called the [[graph of a function|graph]] of $f$.

[[!redirects graph morphisms]]

[[!redirects graph map]]
[[!redirects graph maps]]


graph of a function

#Contents#
* table of contents
{:toc}

## Idea

The _graph of a function_ $f : X \to Y$ is the subset that $f$ "carves out" of the [[cartesian product]] $X \times Y$.

## Definition

### Graph of a function

The traditional standard definition of a graph of a function is this:

+-- {: .un_defn}
###### Definition

The __graph__ $graph(f)$ of a [[function]] $f: X \to Y$ is that [[subset]] $graph(f) \hookrightarrow X \times Y$ of the [[cartesian product]] $X \times Y$ defined by the property that $(a,b) \in X \times Y$ belongs to the graph of $f$ if and only if $f(a) = b$:

$$
  graph(f) = \{(a,b) \in X \times Y | f(a) = b\}
  \,.
$$

=--

This can be understood as a special case of a [[graph of a functor]] by the following observation

+-- {: .un_lemma }
###### Lemma

For $f : X \to Y$ a [[function]], define a function

$$
  \chi_f : X \times Y \to \{0,1\}
$$

by regarding a [[set]] as a [[0-category]] and a 0-category as a [[(-1)-category]]-[[enriched category]] and then setting

$$
  \chi_f : X \times Y \stackrel{=}{\to} X^{op} \times Y \stackrel{f^{op} \times Id}{\to} Y^{op} \times Y \stackrel{Hom}{\to} \{0,1\}
  \,.
$$

Then $\chi_f$ is the [[characteristic function]] of $graph(f)$ in that the diagram

$$
  \array{
     graph(f) &\to& {*}
     \\
     \downarrow && \downarrow
     \\
     X \times Y &\stackrel{\chi_f}{\to}& \{0,1\}
  }
$$

is a [[pullback]] diagram.

=--

In other words this means that in the context of [[(-1)-category]]-[[enriched category theory]] the graph of a function $f$, regarded as an [[enriched functor]] is the [[category of elements]] of the corresponding [[profunctor]]. More on this at [[graph of a functor]].

+-- {: .un_remark}
###### Remark

It is  easy to identify the properties of those subsets of $X \times Y$ that are the graphs of functions, and $f = g$ if they have the same graph (given $X$ and $Y$).  Consequently, it is common, especially (but not only) in [[material set theory]], to *define* a function from $X$ to $Y$ as such a subset, that is to identify a function with its graph.  On the other hand, from a more categorial foundation, as discussed above, it\'s common to define a [[subset]] to be a [[characteristic function]]!

=--

In [[dependent type theory]], a graph of a function $f:A \to B$ is the type

$$\sum_{x:A} \sum_{y:B} f(x) =_A y$$

### Graph of a binary relation

More generally, we can say that the __graph__ of a [[binary relation]] from $X$ to $Y$ is a subset of $X \times Y$; $(a,b)$ belongs to the graph if and only if $a$ is related to $b$.  (Note that *every* subset of $X \times Y$ defines a unique relation; such a subset is the graph of a function if and only if the relation is both [[functional relation|functional]] and [[entire relation|entire]].)

Notice that with a function $f : X \to Y$ regarded as a [[profunctor]] $X \times Y \to (-1)Cat$ as described above, a relation $R \subset X \times Y$ corresponds to a general such [[profunctor]]. More precisely we have a [[pullback]] square

$$
  \array{
    R &\to& {*}
    \\
    \downarrow && \downarrow
    \\
    X \times Y &\stackrel{\chi_R}{\to}& \{0,1\}
  }
$$

where

* $R \subset X \times Y$ is the relation $R$ regarded as a subset of $X \times Y$ in the traditional sense;

* $\chi_R : X \times Y \to \{0,1\}$ is the [[characteristic function]] of this subset.

So in this sense the ordinary notion of relation as a subset does really define the _graph of the relation_, while the relation itself is more naturally understood as the corresponding 0-[[profunctor]]/[[characteristic function]] $\chi_f$.

#### Relation to graph theory

The graph of a binary relation from $X$ to $X$ is related to the notion of [[graph]] from [[graph theory]]; more precisely, such relations correspond to directed loop graphs (in the sense defined at [[graph]]) with vertex set $X$, and either can be defined as a subset of $X^2$.  In a similar way, [[spans]] from $X$ to $X$ correspond to [[digraph|directed pseudographs]] with vertex set $X$.

For the case of a relation from $X$ to $Y$ without $X = Y$, see under the cograph below.

### Graph of an $n$-ary relation

The __graph__ of a [[relation]] of arbitrary arity is similarly a subset of an arbitrary [[cartesian product]]; see [[relation theory]] for more on this.

### Cograph of a function
  {#cograph}

[[Bill Lawvere]] has also considered the __cograph__ of a function, which is dually a [[quotient set]] of the [[disjoint union]] $X \uplus Y$; $a$ is identified with $b$ if $f(a) = b$ (and additional identifications may follow).  However it may make more sense to define the cograph to be a quotient [[poset]] of (the discrete poset) $X \uplus Y$; we declare $a \lt b$ if $f(a) = b$ (and *no* additional relationships follow).  By regarding again a set as a [[0-category]], the latter notion of cograph is a special case of the notion of [[cograph of a functor]], as follows:

A function $f : X \to Y$ determines a [[functor]] $\bar f : I \to Set$ from the [[interval category]] $I = \mathbf{2} = \{a \to b\}$ to [[Set]] by setting $\bar f(a) = X$, $\bar f(b) = Y$ and $\bar f(a \to b) = f$.

Then let $cograph(f)$ be the corresponding [[category of elements]], given by the [[2-pullback]]

$$
  \array{
     cograph(f) &\to& {*}
     \\
     \downarrow && \downarrow
     \\
     I &\stackrel{\bar f}{\to}& Set
  }
$$

which is computed by the strict [[pullback]]

$$
  \array{
     cograph(f) &\to& Set_{*}
     \\
     \downarrow && \downarrow
     \\
     I &\stackrel{\bar f}{\to}& Set
  }
  \,.
$$

The cograph of $f$ in the sense of Lawvere is the set of [[connected component]]s of this category, i.e. $\pi_0(cograph(f))$.

#### Relation to graph theory

The notion of cograph of a function may be even more related to the sense of [[graph]] in graph theory; although the identifications are not done there, the cograph draws a picture in which any relation (or [[multispan]]) of any arity becomes a directed graph (or directed multigraph) whose vertex set is the disjoint union of the relation\'s domains.  When the vertex set is broken up into a disjoint union in this way, graph theorists study this as _multipartite graphs_; in particular, directed bipartite graphs with vertex set broken up as $X + Y$ correspond precisely to binary relations from $X$ to $Y$.

## Generalization

The notion of _graph of a function_ is a special case of the notion [[graph of a functor]] obtained for functors between [[0-category|0-categories]].

Accordingly, the notion of _cograph of a function_ is a special case of the notion of [[cograph of a functor]].

## Related concepts

* [[graph morphism]]

[[!redirects graph of a relation]]
[[!redirects cograph of a function]]
[[!redirects cograph]]
[[!redirects Cograph]]

[[!redirects graph of a function]]