# Questions tagged [cayley-graphs]

Cayley graphs are graphs obtained from a group $G$ in a such way that vertices are elements of the group and edges are added using some generating set $S\subseteq G$.

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### Is there some higher-dimensional polytope that represents the Rubik's Cube group?

I recently found a Pocket Cube and while trying to find instructions on how to solve it, I got sucked into the whole Rubik maths rabbit hole. I was wondering if the Rubik's group can be represented by ...
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### How would a category theorist describe the Cayley graph of a group w.r.t. a subset?

Background: The question at hand is in line with previous questions of mine, such as: How would a category theorist describe Green's relations? Describing the Wreath product categorically. I ask ...
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### Order 12 group with 3 generators, can I reduce to 2 generators?

I'm just getting back into group theory after studying it quite a few years ago. I ran into a seemingly-simple question as I was getting started, looking for advice. I was looking at the dihedral ...
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### What is the Cayley graph for alternating group A6?

According to ATLAS of Group Representations, the alternating group $A_6$ is a group of order 360 which has presentation $$\langle a,b \mid a^2 = b^4 = (ab)^5 = (ab^2)^5 = 1 \rangle.$$ If we draw its ...
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### Do there exists conditions we can put on two groups which have the same growth rate, so that their Cayley graphs are isomorphic?

Given a finitely generated group $G$ with a generating set $S$, we can define the growth rate function of a group, denote it $\#_{G,S}(n)$. It is clear that two groups having the same growth rate ...
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### Detail in construction on Ramanujan graph by LPS

Consider this paper written by Lubotzky, Phillips and Sarnak (1986) on expander graphs. Below definition 2.2, they let $p,q$ be primes both congruent to $1\mod{4}$. Then they claim that there are $p+1$...
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### 2-generated finite non-Abelian simple groups and the existence of Hamiltonian cycles in their Cayley graph

Given that $G = \langle a, b\rangle$ and that $a$ is an involution, when is it the case that there exists $c, d$ such that $G = \langle c, d\rangle$ and $cd$ is an involution? At present, I am ...
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### Normal covering spaces of the wedge sum of $n$ circles

Exercise 1.31 in Hatcher's Algebraic Topology states the following: Show that the normal covering spaces of $S^1 \vee S^1$ are precisely the graphs that are Cayley graphs of groups with two generators....
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### Why Frucht's Theorem is only true for Finite Groups?

The statement of the Frucht's Theorem as follows: "Every Finite Group is Automorphism Group of some graph." The proof involves a result that the group of color preserving Automorphisms of a ...
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### Automorphisms of Cayley graphs of $\mathbb{Z}$

My goal is to show that there are no finite generating sets $A$ and $B$ such that $\mathrm{Cay}(\mathbb{Z},A)$ is isomorphic to $\mathrm{Cay}(\mathbb{Z}\times \mathbb{Z}_2, B)$. My idea for this is to ...
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### The Hamiltonian cycle in a Cayley graph whose corresponding group has a finite cyclic normal subgroup [closed]

Let $S$ generate a finite group $G$ and $s \in S$ such that $\langle s\rangle \trianglelefteq G$, ${\rm Cay}(G/\langle s\rangle,S)$ has a Hamiltonian cycle. Let $(s_1,s_2, \cdots, s_n)$ be the ...
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### Using PlotGraph command from JupyterViz in MyBinder [closed]

I am executing a program "Newprogram.gap" in GAP file in My Binder. There, I have an output which is an adjacency matrix, $A$ corresponding to a Cayley graph. 1). I need to add the command &...
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### Cayley Graphs 1-factorization 4-cycles

Let $X$ be a connected graph on $2^n$ vertices for $n ≥ 1$. Prove that $X$ is a Cayley graph of $\mathit{Z}\,_n^2$ if and only if X has a $1$-factorization such that the union of any two $1$-factors ...
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