hcju/proofwikips huggingface dataset:
Definition:Kronecker Product
Let $\mathbf A = \sqbrk a_{m n}$ and $\mathbf B = \sqbrk b_{p q}$ be matrices. The '''Kronecker product''' of $\mathbf A$ and $\mathbf B$ is denoted $\mathbf A \otimes \mathbf B$ and is defined as the block matrix: :$\mathbf A \otimes \mathbf B = \begin{bmatrix} a_{11} \mathbf B & a_{12} \mathbf B & \cdots & a_{1n} \mathbf B \\ a_{21} \mathbf B & a_{22} \mathbf B & \cdots & a_{2n} \mathbf B \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} \mathbf B & a_{m2} \mathbf B & \cdots & a_{mn} \mathbf B \end{bmatrix}$ Writing this out in full: :$\mathbf A \otimes \mathbf B = \begin{bmatrix} a_{11} b_{11} & a_{11} b_{12} & \cdots & a_{11} b_{1q} & \cdots & \cdots & a_{1n} b_{11} & a_{1n} b_{12} & \cdots & a_{1n} b_{1q} \\ a_{11} b_{21} & a_{11} b_{22} & \cdots & a_{11} b_{2q} & \cdots & \cdots & a_{1n} b_{21} & a_{1n} b_{22} & \cdots & a_{1n} b_{2q} \\ \vdots & \vdots & \ddots & \vdots & & & \vdots & \vdots & \ddots & \vdots \\ a_{11} b_{p1} & a_{11} b_{p2} & \cdots & a_{11} b_{pq} & \cdots & \cdots & a_{1n} b_{p1} & a_{1n} b_{p2} & \cdots & a_{1n} b_{pq} \\ \vdots & \vdots & & \vdots & \ddots & & \vdots & \vdots & & \vdots \\ \vdots & \vdots & & \vdots & & \ddots & \vdots & \vdots & & \vdots \\ a_{m1} b_{11} & a_{m1} b_{12} & \cdots & a_{m1} b_{1q} & \cdots & \cdots & a_{mn} b_{11} & a_{mn} b_{12} & \cdots & a_{mn} b_{1q} \\ a_{m1} b_{21} & a_{m1} b_{22} & \cdots & a_{m1} b_{2q} & \cdots & \cdots & a_{mn} b_{21} & a_{mn} b_{22} & \cdots & a_{mn} b_{2q} \\ \vdots & \vdots & \ddots & \vdots & & & \vdots & \vdots & \ddots & \vdots \\ a_{m1} b_{p1} & a_{m1} b_{p2} & \cdots & a_{m1} b_{pq} & \cdots & \cdots & a_{mn} b_{p1} & a_{mn} b_{p2} & \cdots & a_{mn} b_{pq}  \end{bmatrix}$ Thus, if: :$\mathbf A$ is a matrix with order $m \times n$ :$\mathbf B$ is a matrix with order $p \times q$ then $\mathbf A \otimes \mathbf B$ is a matrix with order $m p \times n q$.
Definition:Kronecker Sum
Let $\mathbf A = \sqbrk a_n$ and $\mathbf B = \sqbrk b_m$ be square matrices with orders $n$ and $m$ respectively. The '''Kronecker sum''' of $\mathbf A$ and $\mathbf B$ is denoted $\mathbf A \oplus \mathbf B$ and is defined as: :$\mathbf A \oplus \mathbf B = \paren {\mathbf A \otimes \mathbf I_m} + \paren {\mathbf I_n \otimes \mathbf B}$ where: :$\otimes$ denotes the Kronecker product :$+$ denotes conventional matrix entrywise addition :$\mathbf I_m$ and $\mathbf I_n$ are the unit matrices of order $m$ and $n$ respectively.
Definition:Column Matrix
A '''column matrix''' is an $m \times 1$ matrix: :$\mathbf C = \begin {bmatrix} c_{1 1} \\ c_{2 1} \\ \vdots \\ c_{m 1} \end {bmatrix}$ That is, it is a matrix with only one column.
Definition:Matrix Direct Sum
Let $\mathbf A = \sqbrk a_{m n}$ and $\mathbf B = \sqbrk b_{p q}$ be matrices. The '''matrix direct sum''' of $\mathbf A$ and $\mathbf B$ is denoted $\mathbf A \oplus \mathbf B$ and is defined as: :$\mathbf A \oplus \mathbf B := \begin {bmatrix} \mathbf A & \mathbf 0 \\ \mathbf 0 & \mathbf B \end {bmatrix}$ where $\mathbf 0$ is a zero matrix, the upper-right $\mathbf 0$ being $m \times q$ and the lower left $\mathbf 0$ being $n \times p$. Thus, if: :$\mathbf A$ is a matrix with order $m \times n$ :$\mathbf B$ is a matrix with order $p \times q$ then $\mathbf A \oplus \mathbf B$ is a matrix with order $\paren {m + p} \times \paren {n + q}$.
Definition:Matrix Addition
=== Matrix Entrywise Addition === This is the usual operation when '''matrix addition''' is specified without qualification): {{:Definition:Matrix Entrywise Addition}} === Matrix Direct Sum === {{:Definition:Matrix Direct Sum}} === Kronecker Sum === {{:Definition:Kronecker Sum}}
Definition:Matrix Product
=== Matrix Product (Conventional) === {{:Definition:Matrix Product (Conventional)}} === Matrix Scalar Product === {{:Definition:Matrix Scalar Product}} === Commutative Matrix Product === {{:Definition:Commutative Matrix Product}} === Kronecker Product === Also known as matrix direct product: {{:Definition:Kronecker Product}} === Hadamard Product === Also known as Matrix Entrywise Product or Schur Product: {{:Definition:Hadamard Product}} === Frobenius Inner Product === {{:Definition:Frobenius Inner Product}} === Cracovian === {{:Definition:Cracovian}}
Definition:Graph (Graph Theory)/Order
Let $G = \struct {V, E}$ be a graph. The '''order''' of $G$ is the cardinality of its vertex set.
Definition:Graph (Graph Theory)/Size
Let $G = \struct {V, E}$ be a graph. The '''size''' of $G$ is the count of its edges.
Definition:Directed Graph
{{:Definition:Directed Graph/Formal Definition}}
Definition:Directed Graph/Arc
Let $G = \struct {V, E}$ be a digraph. The '''arcs''' are the elements of $E$. Informally, an '''arc''' is a line that joins one vertex to another.
Definition:Network
A '''network''' $N = \struct {G, w}$ is: :a graph or digraph $G = \struct {V, E}$ together with: :a mapping $w: E \to \R$ from the edge set $E$ of $G$ into the set $\R$ of real numbers. === Weight Function === {{:Definition:Network/Weight Function}} A general '''network''' can be denoted $N = \struct {V, E, w}$ where the elements are understood to be expressed in the order: vertex set, edge set, weight function. === Weight === {{:Definition:Network/Weight}}
Definition:Edge Set
Let $G = \struct {V, E}$ be a graph. The set $E$ of edges in $G$ is called the '''edge set'''. It consists of (unordered) pairs of elements of the vertex set $V$. It is often convenient to refer to the '''edge set''' for a given graph $G$ as $\map E G$, especially if there is at any one time more than one graph under consideration.
Definition:Loop (Graph Theory)
Let $G = \struct {V, E}$ be a loop-graph. A '''loop''' is an edge $e$ of $G$ whose endvertices are the same vertex. Thus a '''loop''' $e$ on the vertex $v$ would be written: :$e = vv$
Definition:Loop-Graph/Loop-Digraph
A '''loop-digraph''' is a directed graph which allows an arc to start and end at the same vertex: :380px
Definition:Signed Graph
A '''signed graph''' is an undirected network whose functional values are $\pm 1$. The edges of such a graph are known as '''positive edges''' and '''negative edges'''. It is usual to indicate the functional values by means of just the $+$ or $-$ sign. === Example === :320px Another way to represent a '''signed graph''' is to use two different line styles: one to represent '''positive edges''' and one for the '''negative edges'''. Thus the above example can be rendered as: :320px
Definition:Vertex Set
Let $G = \struct {V, E}$ be a graph. The set $V$ of vertices in $G$ is called the '''vertex set'''. It is often convenient to refer to the '''vertex set''' for a given graph $G$ as $\map V G$, especially if there is at any one time more than one graph under consideration.
Definition:Regular Graph
Let $G = \struct {V, E}$ be an simple graph whose vertices all have the same degree $r$. Then $G$ is called '''regular of degree $r$''', or '''$r$-regular'''.
Definition:Complete Graph
Let $G = \struct {V, E}$ be a simple graph such that every vertex is adjacent to every other vertex. Then $G$ is called '''complete'''. The '''complete graph''' of order $p$ is denoted $K_p$.
Definition:Relation Isomorphism
Let $\struct {S_1, \RR_1}$ and $\struct {S_2, \RR_2}$ be relational structures. Let there exist a bijection $\phi: S_1 \to S_2$ such that: :$(1): \quad \forall \tuple {s_1, t_1} \in \RR_1: \tuple {\map \phi {s_1}, \map \phi {t_1} } \in \RR_2$ :$(2): \quad \forall \tuple {s_2, t_2} \in \RR_2: \tuple {\map {\phi^{-1} } {s_2}, \map {\phi^{-1} } {t_2} } \in \RR_1$ Then $\struct {S_1, \RR_1}$ and $\struct {S_2, \RR_2}$ are '''isomorphic''', and this is denoted $S_1 \cong S_2$. The function $\phi$ is called a '''relation isomorphism''', or just an '''isomorphism''', from $\struct {S_1, \RR_1}$ to $\struct {S_2, \RR_2}$.
Definition:Separation (Topology)
Let $T = \struct {S, \tau}$ be a topological space. Let $A$ and $B$ be open sets of $T$. $A$ and $B$ form a '''separation of $T$''' : :$(1): \quad A$ and $B$ are non-empty :$(2): \quad A \cup B = S$ :$(3): \quad A \cap B = \O$
Definition:Component of Graph
Let $G$ be a graph. Let $H$ be a subgraph of $G$ such that: :$H$ is connected :$H$ is not contained in any connected subgraph of $G$ which has more vertices or edges than $H$ has. Then $H$ is a '''component''' of $G$.
Definition:Cut-Vertex
Let $G = \struct {V, E}$ be a connected graph. Let $v$ be a vertex of $G$. Then $v$ is a '''cut-vertex''' of $G$  the vertex deletion $G - v$ is a vertex cut of $G$. That is, such that $G - v$ is disconnected. Thus, a '''cut-vertex''' of $G$ is a singleton vertex cut of $G$.
Definition:Bridge
Let $G = \struct {V, E}$ be a connected graph. Let $e \in E$ be an edge of $G$ such that the edge deletion $G - e$ is disconnected. Then $e$ is known as a '''bridge''' of $G$.
Definition:Semi-Eulerian Graph
A graph is called '''semi-Eulerian''' if it contains an Eulerian trail. Note that the definition of graph here includes: * Simple graph * Loop-graph * Multigraph * Loop-multigraph. Note that an Eulerian graph is also '''semi-Eulerian''', as an Eulerian circuit is still a path, and therefore an Eulerian trail.
Definition:Eulerian Trail
An '''Eulerian trail''' is a trail $T$ that passes through every vertex of a graph $G$ and uses every edge of $G$ exactly once.
Definition:Loop-Graph
A '''loop-graph''' is a graph which allows an edge to start and end at the same vertex:
Definition:Multigraph/Multiple Edge
Let $G = \struct {V, E}$ be a multigraph. A '''multiple edge''' is an edge of $G$ which has another edge with the same endvertices. That is, where there is more than one edge that joins any pair of vertices, each of those edges is called a multiple edge.
Definition:Edge-Disjoint Trails
Let $G = \struct {V, E}$ be an undirected graph. Let $T_1$ and $T_2$ be trails in $G$. Then $T_1$ and $T_2$ are '''edge-disjoint'''  they have no edges in common.
Definition:Degree Sequence
Let $G = \struct {V, E}$ be a graph. The '''degree sequence''' of $G$ is the sequence obtained by listing all the degrees of all the vertices of $G$ in ascending order, with repeats as needed.
Definition:Edgeless Graph
An '''edgeless graph''' is a graph with no edges. That is, an '''edgeless graph''' is a graph of size zero. Equivalently, an '''edgeless graph''' is a graph whose vertices are all isolated. The '''edgeless graph''' of order $n$ is denoted $N_n$ and can be referred to as the '''$n$-edgeless graph'''.
Definition:Cycle Graph
A '''cycle graph''' is a graph which consists of a single cycle. The cycle graph of order $n$ is denoted $C_n$. A cycle graph is $2$-regular.
Definition:Cubic Graph
A '''cubic graph''' is a $3$-regular graph, that is, a graph whose vertices all have degree $3$.
Definition:Platonic Graph
The '''Platonic graphs''' are a set of graphs representing the edges and vertices of the five Platonic solids. === Tetrahedron === {{:Definition:Platonic Graph/Tetrahedron}} === Octahedron === {{:Definition:Platonic Graph/Octahedron}} === Cube === {{:Definition:Platonic Graph/Cube}} === Icosahedron === {{:Definition:Platonic Graph/Icosahedron}} === Dodecahedron === {{:Definition:Platonic Graph/Dodecahedron}}
Definition:Graph (Graph Theory)/Edge
Let $G = \struct {V, E}$ be a graph. The '''edges''' are the elements of $E$.
Definition:Petersen Graph
The '''Petersen graph''' is a $3$-regular graph of order $10$ and size $15$, as follows: :300px {{NamedforDef|Julius Petersen|cat=Petersen}}
Definition:Bipartite Graph
A '''bipartite graph''' is a graph $G = \left({V, E}\right)$ where: :$V$ is partitioned into two sets $A$ and $B$ such that: :each edge is incident to a vertex in $A$ and a vertex in $B$.
Definition:Complete Bipartite Graph
A '''complete bipartite graph''' is a bipartite graph $G = \left({A \mid B, E}\right)$ in which every vertex in $A$ is adjacent to every vertex in $B$. The complete bipartite graph where $A$ has $m$ vertices and $B$ has $n$ vertices is denoted $K_{m, n}$. Note that $K_{m, n}$ is the same as $K_{n, m}$.
Definition:Path Graph
A '''path graph''' is a tree which has a path which passes through all its vertices. The path graph with $n$ vertices is denoted $P_n$.
Definition:Hamiltonian Path
A '''Hamiltonian path''' in a graph is a path (not a cycle) that contains every vertex of the graph (but not necessarily every edge).
Definition:Semi-Hamiltonian Graph
A '''semi-Hamiltonian graph''' is a graph that contains a Hamiltonian path, but ''not'' a Hamilton cycle.
Definition:Kolmogorov Space
{{:Definition:Kolmogorov Space/Definition 1}}
Definition:Balanced Signed Graph
Let $G = \left({V, E}\right)$ be a signed graph. Then $G$ is '''balanced''' if it is possible to partition $V$ into two subsets $A, B$ such that: * the positive edges have both ends in $A$ or both ends in $B$; * the negative edges have one end in $A$ and the other end $B$. It is a frequent practice to apply a different colour to the vertices in each set to enhance clarity.
Definition:Forest
A '''forest''' is a simple graph whose components are all trees. A '''connected forest''' is just a single tree.
Definition:Labeled Graph
A '''labeled graph''' is a graph whose vertices are each assigned an element from a set of symbols (letters, usually, but this is unimportant). The important thing to note is that the vertices can be distinguished one from another. Note that it is not necessary for all the vertices to be assigned different labels. A vertex-colored graph can be considered as a labeled graph, in which the labels assigned are elements from a set of colors. In this context it is then rare that all vertices are required to be assigned different colors. A graph which has no such labeling is called an '''unlabeled graph'''.
Definition:Binary Tree
A '''binary tree''' is a rooted tree which has at most two branches at any node. That is, every node in a '''binary tree''' has (apart from the root) degree of either $1$ (for the leaf nodes) or $3$ (one for the parent, two for the children). The branches at any particular node are frequently called the '''left-hand branch''' and the '''right-hand branch''', and a distinction is made between them.
Definition:Prüfer Sequence
A '''Prüfer sequence''' of order $n$ is a (finite) sequence of integers: :$\left({\mathbf a_1, \mathbf a_2, \ldots, \mathbf a_{n-2}}\right)$ such that $\forall i: 1 \le i \le n-2: 1 \le \mathbf a_i \le n$. That is, it is a (finite) sequence of $n - 2$ integers between $1$ and $n$.
Definition:Sierpiński Number of the Second Kind
A '''Sierpiński number of the second kind''' is an odd positive integer $k$ such that integers of the form $k2^n + 1$ are composite for all positive integers $n$. That is, when $k$ is a '''Sierpiński number of the second kind''', all members of the set: :$\left\{{k 2^n + 1}\right\}$ are composite.
Definition:Center (Abstract Algebra)/Ring
The '''center of a ring''' $\struct {R, +, \circ}$, denoted $\map Z R$, is the subset of elements in $R$ that commute with every element in $R$. Symbolically: :$\map Z R = \map {C_R} R = \set {x \in R: \forall s \in R: s \circ x = x \circ s}$ That is, the '''center of $R$''' is the centralizer of $R$ in $R$ itself. It is clear that the '''center of a ring''' $\struct {R, +, \circ}$ can be defined as the center of the group $\struct {R, \circ}$. {{explain|I think the group is not $R$, but $R^*$, to avoid finding an inverse for the zero under multiplication.<br>Taking this question to Discussion.}}
Definition:Center of Tree
Take a tree, and remove all the nodes whose degree is $1$, along with all their incident edges. Repeat the process until either: :One node is left or: :Two nodes are left, joined by a single edge. If one node is left, it is called the '''center''' of the tree. If two nodes are left, joined by a single edge, this is called the '''bicenter''' of the tree. === Central Tree === {{:Definition:Central Tree}} === Bicentral Tree === {{:Definition:Bicentral Tree}}
Definition:Sophie Germain Prime
A '''Sophie Germain prime''' is a prime number $p$ such that $2 p + 1$ is also prime.
Definition:Spanning Tree
Let $G$ be a connected graph. A '''spanning tree for $G$''' is a spanning subgraph of $G$ which is also a tree. {{refactor|Move the below into its own page}} Clearly a tree is its own spanning tree: As a tree $T$ of order $n$ has $n - 1$ edges, its spanning tree must also contain $n-1$ edges, and those must be the same ones as in $T$.
Definition:Minimum Spanning Tree
Let $G$ be a weighted graph. The '''minimum spanning tree''' for $G$ is a spanning tree for $G$ which has a minimum total weight. For a given $G$, the '''minimum spanning tree''' may not be unique. It can also be called the '''minimum connector''' for $G$.
Definition:Maximum Spanning Tree
Let $G$ be a weighted graph. The '''maximum spanning tree''' for $G$ is a spanning tree for $G$ which has a maximum total weight. For a given $G$, the '''maximum spanning tree''' may not be unique. It can also be called the '''maximum connector''' for $G$.
Definition:Greedy Algorithm
A '''greedy algorithm''' is an algorithm whose decision strategy is such that any choice made at any stage takes no consideration of any future state. The strategy is to make the choice which has the greatest short-term effect towards the long-term goal. In some problems this may not produce the optimum solution.
Definition:Adjacency Matrix
An '''adjacency matrix''' is a matrix which describes a graph by representing which vertices are adjacent to which other vertices. If $G$ is a graph of order $n$, then its '''adjacency matrix''' is a square matrix of order $n$, where each row and column corresponds to a vertex of $G$. The element $a_{i j}$ of such a matrix specifies the number of edges from vertex $i$ to vertex $j$. An '''adjacency matrix''' for a simple graph and a loop-digraph is a logical matrix, that is, one whose elements are all either $0$ or $1$. An '''adjacency matrix''' for an undirected graph is symmetrical about the main diagonal. This is because if vertex $i$ is adjacent to vertex $j$, then $j$ is adjacent to $i$. An '''adjacency matrix''' for a weighted graph or network contains the weights of the edges.
Definition:Witch of Agnesi
:620px Let $OAM$ be a circle of radius $a$ whose center is at $\tuple {0, a}$. Let $M$ be the point such that $OM$ is a diameter of $OAM$. Let $OA$ be extended to cut the tangent to the circle through $M$ at $N$. Generate $NP$ perpendicular to $MN$ and $AP$ parallel to $MN$. As $A$ moves around the circle $OAM$, the point $P$ traces the curve known as the '''Witch of Agnesi'''.
Definition:Preordering
Let $\mathcal R \subseteq S \times S$ be a relation on a set $S$. === Definition 1 === {{:Definition:Preordering/Definition 1}} === Definition 2 === {{:Definition:Preordering/Definition 2}}
Definition:Partial Preordering
Let $S$ be a set. Let $\precsim$ be a preordering on $S$. Then $\precsim$ is a '''partial preordering''' on $S$ iff $\precsim$ is not connected. That is, iff there is at least one pair of elements of $S$ which is non-comparable: :$\exists x, y \in S: x \not \precsim y \land y \not \precsim x$
Definition:Total Preordering
Let $S$ be a set. Let $\precsim$ be a preordering on $S$. Then $\precsim$ is a '''total preordering''' on $S$  $\precsim$ is connected. That is,  there is no pair of elements of $S$ which is non-comparable: :$\forall x, y \in S: x \precsim y \lor y \precsim x$
Definition:Trichotomy
Let $S$ be a set. A '''trichotomy''' on $S$ is a relation $\RR$ on $S$ such that for every pair of elements $a, b \in S$, '''exactly''' one of the following three conditions applies: :$a \mathrel \RR b$ :$a = b$ :$b \mathrel \RR a$
Definition:Right-Total Relation
Let $S$ and $T$ be sets. Let $\mathcal R \subseteq S \times T$ be a relation in $S$ to $T$. Then $\mathcal R$ is '''right-total''' : :$\forall t \in T: \exists s \in S: \tuple {s, t} \in \mathcal R$ That is,  every element of $T$ is related to by some element of $S$. That is, : :$\Img {\mathcal R} = T$ where $\Img {\mathcal R}$ denotes the image of $\mathcal R$.
Definition:Left-Total Relation
Let $S$ and $T$ be sets. Let $\mathcal R \subseteq S \times T$ be a relation in $S$ to $T$. Then $\mathcal R$ is '''left-total''' : :$\forall s \in S: \exists t \in T: \tuple {s, t} \in \mathcal R$ That is,  every element of $S$ relates to some element of $T$.
Definition:Endorelation
Let $S \times S$ be the cartesian product of a set or class $S$ with itself. Let $\RR$ be a relation on $S \times S$. Then $\RR$ is referred to as an '''endorelation on $S$'''.
Definition:Correspondence
A relation $\RR \subseteq S \times T$ is a '''correspondence'''  $\RR$ is both left-total and right-total.
Definition:Coreflexive Relation
{{:Definition:Coreflexive Relation/Definition 1}}
Definition:Reflexive Relation
{{:Definition:Reflexive Relation/Definition 1}}
Definition:Antireflexive Relation
$\RR$ is '''antireflexive''' : :$\forall x \in S: \tuple {x, x} \notin \RR$
Definition:Symmetric Relation
{{:Definition:Symmetric Relation/Definition 1}}
Definition:Asymmetric Relation
{{:Definition:Asymmetric Relation/Definition 1}}
Definition:Antisymmetric Relation
{{:Definition:Antisymmetric Relation/Definition 1}}
Definition:Transitive Relation
{{:Definition:Transitive Relation/Definition 1}}
Definition:Antitransitive Relation
$\mathcal R$ is '''antitransitive''' : :$\left({x, y}\right) \in \mathcal R \land \left({y, z}\right) \in \mathcal R \implies \left({x, z}\right) \notin \mathcal R$ that is: : $\left\{ {\left({x, y}\right), \left({y, z}\right)}\right\} \subseteq \mathcal R \implies \left({x, z}\right) \notin \mathcal R$
Definition:Euclidean Relation
Let $\mathcal R \subseteq S \times S$ be a relation in $S$.
Definition:Serial Relation
Let $\mathcal R \subseteq S \times S$ be a relation in $S$. $\mathcal R$ is '''serial''' : :$\forall x \in S: \exists y \in S: \tuple {x, y} \in \mathcal R$
Definition:Totally Ordered Ring
Let $\struct {R, +, \circ, \preceq}$ be an ordered ring. If the ordering $\preceq$ is a total ordering, then $\struct {R, +, \circ, \preceq}$ is a '''totally ordered ring'''.
Definition:Isolated Vertex
If the degree of $v$ is zero, then $v$ is an '''isolated vertex'''.
Definition:Locally Finite Graph
A '''locally finite graph''' $G$ is an infinite graph where every vertex of $G$ has finite degree.
Definition:Infinite Graph
A graph which has either an infinite number of edges or vertices is called an '''infinite graph'''.
Definition:Graph (Graph Theory)/Edge/Endvertex
Let $G = \struct {V, E}$ be a graph or digraph. Let $e = u v$ be an edge of $G$, that is, $e \in E$. The '''endvertices''' of $e$ are the vertices $u$ and $v$.
Definition:Link (Graph Theory)
Let $G = \left({V, E}\right)$ be a graph. A '''link''' is a edge of $G$ which is ''not'' a loop. That is, a '''link''' is an edge whose endvertices are  distinct. When $G$ is not a loop-graph, then ''all'' edges are '''links'''.
Definition:Complement (Graph Theory)
=== Simple Graph === {{:Definition:Complement (Graph Theory)/Simple Graph}} === Loop-Graph === {{:Definition:Complement (Graph Theory)/Loop-Graph}}
Definition:Null Graph
The '''null graph''' is the graph which has no vertices. That is, the '''null graph''' is the graph of order zero. It is called '''''the'' null graph''' because, from Empty Set is Unique, there is only one such entity.
Definition:Group Homomorphism
Let $\struct {G, \circ}$ and $\struct {H, *}$ be groups. Let $\phi: G \to H$ be a mapping such that $\circ$ has the morphism property under $\phi$. That is, $\forall a, b \in G$: :$\map \phi {a \circ b} = \map \phi a * \map \phi b$ Then $\phi: \struct {G, \circ} \to \struct {H, *}$ is a group homomorphism.
Definition:Ring Homomorphism
Let $\struct {R, +, \circ}$ and $\struct{S, \oplus, *}$ be rings. Let $\phi: R \to S$ be a mapping such that both $+$ and $\circ$ have the morphism property under $\phi$. That is, $\forall a, b \in R$: {{begin-eqn}} {{eqn | n = 1       | l = \map \phi {a + b}       | r = \map \phi a \oplus \map \phi b }} {{eqn | n = 2       | l = \map \phi {a \circ b}       | r = \map \phi a * \map \phi b }} {{end-eqn}} Then $\phi: \struct {R, +, \circ} \to \struct {S, \oplus, *}$ is a ring homomorphism.