Questions tagged [cayley-graphs]

Questions concerning Cayley graphs, regardless of whether the group be finite, infinite, abelian, non-abelian. Strong connections to geometric group theory.

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Perfect Cayley graphs for abelian groups have $\frac{n}{\omega}$ disjoint maximal cliques

Let $G$ be a perfect/ weakly perfect Cayley graph on an abelian group with respect to a symmetric generating set. In addition let the clique number be $\omega$ which divides the order of graph $n$. ...
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1answer
130 views

Cayley graphs do not have isolated maximal cliques

Let a Cayley graph $G$ of a group $H$ with respect to the generating set $\{s_i\}$ have a clique of order $> 2$. In addition assume the graph $G$ is non-complete. If the clique size is less than ...
5
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2answers
263 views

Transposition Cayley graphs are planar

Consider the Cayley graph $G$ with vertex set the elements of the symmetric group $S_n$ and generating set the set of minimal transposition generators of the group $S_n$, that is the set $S=\{(12),(13)...
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1answer
44 views

Recognizing perfect cayley graphs as tensor products

It is known (and can easily be seen) that a unitary cayley graph on $n=\prod_ip_i$, ($p_i$ prime) vertices with $n$ squarefree can be recognized as the tensor product of the graphs $K_{p_i}$, where $...
4
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1answer
211 views

$C_4\times C_2 : C_2$: what does this mean?

I am reading this paper where the object $C_4\times C_2 : C_2$ is used as a group structure. I know that $C_n$ is a cyclic group but don't know what kind of operation between groups is identified by ...
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0answers
91 views

Chromatic number of certain graphs with high maximum degree

Let $G$ be the graph of even order $n$ and size $\ge\frac{n^2}{4}$ which is a Cayley graph on a nilpotent group but not complete. Can the chromatic number of this graph be determined in polynomial ...
3
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1answer
179 views

Diameter of Cayley graphs of finite simple groups

Babai, Kantor and Lubotzky proved in 1989 the following theorem (Sciencedirect link to article). THEOREM 1.1. There is a constant $C$ such that every nonabelian finite simple group $G$ has a set $S$ ...
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1answer
84 views

A vertex transitive graph has a near perfect/ matching missing an independent set of vertices

Consider a power of cycle graph $C_n^k\,\,,\frac{n}{2}>k\ge2$, represented as a Cayley graph with generating set $\{1,2,\ldots, k,n-k,\ldots,n-1\}$ on the Group $\mathbb{Z}_n$. Supposing I remove ...
6
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1answer
204 views

Vanishing of certain coefficients coming from Coxeter groups

Let $\left(W\text{, }S\right)$ be a Coxeter system. For every $w\in W$ let us write $\left|w\right|$ for the length of $w$. Set $\lambda\left(e\right)=1$ where $e\in W$ denotes the neutral element of ...
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0answers
124 views

Are all even regular undirected Cayley graphs of Class 1?

Are even order Cayley graphs of Class 1, that is, can they be edge-colored with exactly $m$ colors, where $m$ is the degree of each vertex? I think yes, because of the symmetry the Cayley graphs ...
4
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1answer
123 views

Diameter for permutations of bounded support

Let $S\subset \textrm{Sym}(n)$ be a set of permutations each of which is of bounded support, that is, each $\sigma\in S$ moves $O(1)$ elements of $\{1,2,\dotsc,n\}$. Let $\Gamma$ be the graph whose ...
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0answers
161 views

Growth functions of finite group - computation, typical behaviour, surveys?

Looking on the growth function for Rubik's group and symmetric group, one sees rather different behaviour: Rubik's growth in LOG scale (see MO322877): S_n n=9 growth and nice fit by normal ...
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1answer
118 views

Total coloring conjecture for Cayley graphs

The total coloring conjecture (TCC) states that any total coloring of a simple graph $G(V,E)$ has its total chromatic number bounded as $\chi^{T}(G)\le \Delta+2$ where $\Delta $ is the maximal degree ...
3
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1answer
111 views

Cliques in Cayley graph on $n$-cycles

Let $S\subset S_n$ be the set of all $n$-cycles. I want to know if the Cayley graph $(S_n,S)$ has large dense subgraphs. I'm expecting it to not have super-polynomial size and $1-o(1)$ dense subgraphs....
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1answer
299 views

Cayley graph properties

Consider an infinite graph that satisfies the following property: if any finite set of vertices is removed (and all the adjacent edges), then the resulting graph has only one infinite connected ...
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1answer
359 views

Chromatic numbers of infinite abelian Cayley graphs

The recent striking progress on the chromatic number of the plane by de Grey arises from the interesting fact that certain Cayley graphs have large chromatic number; namely, the graph whose vertices ...
2
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2answers
137 views

Distance regular Cayley graphs on $Z_2^n$?

Let $Z_2^n$ be group $Z_2 \times Z_2 \times \cdots \times Z_2$ with operation Exclusive-or. I'd like to know if the $Cay(Z_2^n,S)$ for $S \subset Z_2^n \setminus \{0\}$ is distance regular graph or ...
18
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4answers
833 views

Cayley graph of $A_5$ with generators $(1,2,3,4,5),(1,4,3,2,5)$

The Cayley graph of $A_5$ with two generators of order 5 seems rather complicated. What is its graph genus (orientable or non-orientable)? The best I could get by trial and error is an embedding ...
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2answers
111 views

coloring infinite vertex transitive graph without large cliques

Let $G$ be an infinite vertex-transitive graph (this means that for every $u,w \in V(G)$ there exists an automorphism $\tau$ of $G$ such that $\tau(u) = v$). We assume that $G$ is undirected, and does ...
5
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1answer
118 views

Inertia of a class of Cayley graphs

Let $H^n_2(d)$ be the Cayley graph with vertex set $\{0,1\}^n$ where two strings form an edge iff they have Hamming distance at least $d$. What is the inertia of these graphs, that is, the numbers of ...
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2answers
512 views

A generously vertex transitive graph which is not Cayley?

A graph is vertex transitive if $x \mapsto y$ by an automorphism. A graph is generously vertex transitive if $x \mapsto y \mapsto x$ by an automorphism. Simple facts: GVT $\rightarrow$ unimodular. ...
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0answers
200 views

Approximation of the effective resistance on Cayley graph

Let $\Gamma$ be a finitely generated group, and denote by $G$ the Cayley graph of $\Gamma$. Denote by $d_R$ the resistance distance metric on this graph. The resistance distance metric between the ...
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0answers
271 views

Induced graphs of Cayley graph

I have a Cayley graph $\mathrm{Cay}(G,S)$, its group presentation $G=\langle S | R \rangle$, and it becomes a metric graph by assigning a length equal to $1$ to each edge. I also have an induced ...
5
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0answers
250 views

Complexity of property of being vertex-transitive resp. of being a Cayley graph

Suppose I gave you a finite graph, and asked you whether it was vertex-transitive. How hard is that algorithmically? The second question is: suppose I gave you a vertex transitive finite graph, and ...
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4answers
1k views

Is there a Cayley graph of a non-abelian finite group that is not isomorphic to any Cayley graph of any abelian group?

It's the first question I post here :) I'm sorry if the question is too specific or if it's somehow repeating others. In other words, my question is the following. Consider a Cayley graph $\Gamma$ of ...
25
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4answers
1k views

What relationship, if any, is there between the diameter of the Cayley graph and the average distance between group elements?

It's known that every position of Rubik's cube can be solved in 20 moves or less. That page includes a nice table of the number of positions of Rubik's cube which can be solved in k moves, for $k = 0,...
7
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2answers
1k views

Quasi-isometries vs Cayley Graphs

The following questions might be trivial, however, I couldn't solve them: Let $G$ be generated by a finite symmetric set $S$. Suppose that $\Gamma(G,S)$ is the corresponding right Cayley graph of $G$...
14
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6answers
2k views

Spectral properties of Cayley graphs

Let $G$ be a finite group. Do eigenvalues of its Cayley graph say anything about the algebraic properties of $G$? The spectrum of Cayley graph may depend on the presentation, so it's not a good ...
68
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6answers
7k views

Which graphs are Cayley graphs?

Every group presentation determines the corresponding Cayley graph, which has a node for each group element, and arrows labeled with the generators to get from one group element to another. My main ...