Talk:Cayley graph

message: What is the history of the Cayley graph?  Did Arthur Cayley have anything to do with it?  How about some theorems that use the notion?  --Dbenbenn 05:46, 11 Dec 2004 (UTC)
thread_title:  History 

message: Mathematics Manual of Style recommends that inline math be used for very simple formulas, and it's now almost a universal practice for mathematics articles. Here most sections are still written using latex markup (I preserved the style in the section that I've just edited), but it seems like an overkill and I propose to uniformly switch to inline. Arcfrk (talk) 01:49, 29 February 2008 (UTC)
thread_title:  TeX vs inline math 

message: I've removed the following text from the article. I am unfamiliar with the use of the term "Bruhat order" outside of the context of Coxeter groups, and in any case, there is very little of value for the topic of Cayley graphs (saying "minimal path" is the same as "reduced word" is the same as "minimal path" is nonsensical). Arcfrk (talk) 23:30, 11 April 2009 (UTC)
thread_title:  Moved from the article (OR) 

message: There are several issues about this page from the point of view of algebraic graph theory.
# The definition produces a graph with coloured edges, but in algebraic graph theory the edges are almost always left uncoloured unless the name "Cayley colour graph" is used (which is rare).  This convention can be seen also on this page with the theorem "A graph Γ is a Cayley graph of a group G if and only if it admits a simply transitive action of G by graph automorphisms."  If the definition on this page was applied rigorously, Γ would need to be edge-coloured in a particular way before this theorem is true.  However, the uncoloured version is true and less trivial.
# The page requires Cayley graphs to be connected, but in algebraic graph theory the disconnected case is usually allowed too.
Suggestions for fixing this are solicited. McKay (talk) 07:58, 9 May 2009 (UTC)
replies: text: So in the disconnected case, one has a group and a set of group elements that is not necessarily a generating set? Or a groupoid and a generating set for it? —David Eppstein (talk) 15:30, 9 May 2009 (UTC)
replies: text:  The former.  For the directed case, a group G and a set of elements S of the group.  In the undirected case, S must also be closed under inverse.  In both cases, many authors disallow the identity element from being in  S. Excluding the case where S doesn't generate G causes more graph-theoretic trouble than it's worth, for example it means that the complement of a Cayley graph may or may not be a Cayley graph, the tensor (direct) product of two Cayley graphs may or may not be a Cayley graph, etc. McKay (talk) 03:07, 10 May 2009 (UTC)

text: No objections from me. —David Eppstein (talk) 04:17, 10 May 2009 (UTC)
thread_title:  terminology of this page 

message: When reading this article, I was somewhat surprised to see that although Cayley graphs are referred to as begin "a central tool in combinatorial and geometric group theory", no reference is made to the association with the Word Problem, one of the central problems in these areas! So I have added in a quick line about this in what I deemed to be a somewhat appropriate part of the article. I appologise for the sloppy editing, it is the first time I have referenced something on here!
I have also added a few alternative names for the Cayley graph which have been used in the literature (for instance, color group was used by both Cayley and Burnside).  —Preceding unsigned comment added by 130.209.6.40 (talk) 09:42, 22 April 2010 (UTC)
thread_title:  Word Problem 

message: The article right now (17:47, 19 October 2010 (UTC)
replies: text: The trouble is that a simplified explanation will lack formal mathematical rigor, and these are complicated and subtle concepts.  A Cayley graph is a representation of a discrete group.  This is a concept that requires subtle understanding of continuity and discreteness (as well as formal definitions of graphs and groups).
Now, it would be rather hard to understand the definition of a Cayley graph for a student who does not have a good grasp of elementary ("naive") set theory. Just "understanding" graphs and groups by means of familiarising yourself with some examples (which is what school pupils often do, in lieu of assimilating formal definitions), is hardly enough in this case. This indeed may make the Cayley graphs a bit hard to digest for eight graders (even bright ones). However, there is no need whatsoever to learn about topology in general or topological groups in particular, in order to grasp the definition; and certainly no need to learn the differences between e.g. discreteness and connectedness for Hausdorff spaces, for a first understanding of Cayley graphs.
IMHO, the formulation "discrete group" thus is fooling not just eighth graders, but also reference desk answererers, getting them to believe that the concept is more complicated than it is. I'll remove it, and suggest that if it is reinstated the information that the group may be viewed as having the trivial topology is postponed until it is employed (if it is). JoergenB (talk) 17:47, 19 October 2010 (UTC)
thread_title:  Too much 'discretion' 

message: The definition of the Cayley graph in the Definition section says that we have an edge going from 'g' to 'gs', labeled 's'; so that the edge represents multiplication on the right by the element labeling the edge.
However, the fourth example and the attached graph use the opposite convention, by using a red arrow associated to 'a' to connect 'x' with 'ax'. The text correctly states that the arrow represents multiplication by 'a' on the left (rather than the right), but it is perhaps somewhat confusing to have this change of convention without any "announcement." Perhaps one might add somewhere along the line that one can likewise define the graph using left multiplication instead of right, or to modify the image/description to avoid the issue. Magidin (talk) 22:22, 19 May 2012 (UTC)
replies: text: That is correct, and I have changed it. Five years ago, in the discussion above, the IP has convinced my to change it to prefix notation - but that seems to be unusual in Cayley graphs. There was a Stackexchange discussion BTW, whether or not the prefix notation was wrong: http://math.stackexchange.com/questions/146923/is-the-cayley-diagram-for-d4-in-wikimedia-commons-wrong --Watchduck (quack) 00:54, 28 November 2016 (UTC)
thread_title:  Clash of conventions in definition and example 

message: 77.126.182.100 (talk) 11:36, 2 November 2012 (UTC)
thread_title:  Why does "Cayley diagram" redirect to Cognate_linkage rather than here? 

message: "Suppose that G=\mathbb{Z} \! is the infinite cyclic group and the set S consists of the standard generator 1 and its inverse (−1 in the additive notation) then the Cayley graph is an infinite chain."
no it isn't!!! between n and n+1 there are TWO edges, non one!--93.66.195.134 (talk) 12:26, 5 April 2013 (UTC)
thread_title:  wrong definition of cayley graph 

message: The reference to the connectivity of the graph (Babai's Technical Report TR-94-10) is a dead link.  — Preceding unsigned comment added by 180.48.56.132 (talk) 09:48, 15 March 2015 (UTC)
replies: text: Fixed (replaced by published version). —David Eppstein (talk) 18:00, 15 March 2015 (UTC)
thread_title:  Reference dead link 

message: This is a poor heading for a jumbled collection of properties, conventions and theorems; some of which are anything but elementary, because they assume a knowledgeable reader.  yoyo (talk) 16:22, 13 December 2018 (UTC)
thread_title:  Elementary properties 

message: I reworked the recently added section on expanding Cayley graphs, which contained some wrong statements, most notably about the relation between property (T) and expansion. Prperyy (T) is not equivalent to expansion of the Cayley graphs, for example free groups have an expanding Cayley graph but are not Kazhdan. It is not even equivalent to uniform expansion of all quotients since this condition holds for the non-Kazhdan group SL_2(Z[sqrt 2]). The example of abelian groups is not relevant for an article of which expansion is not the primary topic so i removed it. jraimbau (talk) 13:27, 19 November 2021 (UTC)
thread_title:  Expansion