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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 $S\subseteq G$.

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 $S\subseteq G$.

218 questions
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Petersen graph is not a Cayley graph

How can I show that the Petersen graph is not a Cayley graph? I don't know very much about Cayley graphs, I know that they are vertex-transitive, but so is the Petersen graph. It probably has to do with the group structure of $\Gamma$ in…
hannahh
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Why is the Petersen graph no Cayley graph?

On the Wikipedia page of the Petersen graph it is mentioned that it is not a Cayley graph. How it this proved? Honestly I don't even know how to start this. The only criteria I can think of is that all vertices must have the same degree. Also the…
stacky
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Applications of Cayley Graphs in Physics

I have been recently reading about Cayley graphs and character theory. It is evident that Cayley graphs are very useful tool in theoretical computer science. In physics, Cayley graphs seem do appear in the study of quantum walks. I wonder however,…
Abbas
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What does the Cayley graph of the Grigorchuk group 'look like'?

I've recently renewed my interest in tilings, and as a result have taken some splashes into Word Processing in Groups (in search of good information on the automatic groups related to hyperbolic tilings) and the amazing The Symmetries of Things. …
Steven Stadnicki
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Computing shortest paths in Cayley graphs

I am interested in shortest paths in the Cayley graph of the alternating group $A_{12}$ acting on the vertices of the icosahedron, where the generators are given by 5-cycles on the neighbors of any particular vertex. Is there a decent algorithm for…
Geoffrey Irving
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Are jumps in the growth function of an infinite group increasing?

Let $G$ be a group with a $S$ a finite subset of $G$ generating it, with $\{e\}\in S$ and $S=S^{-1}$, and let $\gamma_G^S$ be the growth function of $G$ respect to $S$, that is, $\gamma_G^S(l)$ is the number of elements of $G$ which can be expressed…
Saúl RM
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Diameter of undirected graph

Let $G$ be a strongly connected directed graph of diameter $D$, and suppose that we remove the orientation of the arcs, thus getting an undirected graph $G'$ with diameter $D'$. Obviously, $D' \leq D$. What else can be said about $D$ and $D'$?. In…
Manolito Pérez
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Informal definition of groups in Nathan Carter's "Visual Group Theory"

Context In Visual Group Theory by Nathan Carter, the author introduces groups by considering the set of all actions generated by a given collection of actions, which are called generators. In section 1.4, the author lays down the conditions that the…
Soham Saha
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The number of closed paths in the square lattice $\mathbb{Z}^2$ with length $n$ and starting and ending points at $(0,0)$.

I'm thinking about this problem right now. Problem:Consider a lattice point consisting of $\mathbb{Z}^2$ points. If $n$ is even, i.e., $n=2p$, then Show that the number of closed paths in the square lattice $\mathbb{Z}^2$ with length $n$ and…
epsilon
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Software for generating Cayley graphs of $\mathbb Z_n$?

Does it exist any program (for linux) which can generate a nice Cayley graph of any $\mathbb Z_n$? (If it's possible to create such a graph at all, that is.) (where perhaps $n ≤ 100$ or something like that)
John Smith
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Is there a ternary Cayley graph on $27$ vertices that is a non-complete core?

I am thankful for feedback given to me, through comments on one question on MSE that I have deleted and also through responses to my question on Meta. With this feedback, I am rewriting another question that I have deleted, adding some context, and…
Colin Tan
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Random walk on the lamplighter group

Consider a random walk on the lamplighter group with the following generating set: move left, move right, and toggle lamp. Start at the origin, with all lamps off. What is the probability that, after $t$ steps, the lamp at the origin is on? I…
user76284
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Hatcher Covering Spaces Ex. 11 & 31 and Surjectivity of the Covering Map

I am confused by the statements of a couple of the exercises in Section 1.3 of Hatcher. I think they need additional hypotheses that are not reflected in Hatcher's errata. Exercise 11: Construct finite graphs $X_1$ and $X_2$ having a common…
Julius
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on Cayley diagrams

is the picture the Cayley Graph of the group $\langle a,b,c\mid a^2, b^2,c^2\rangle$ ? What would it be for $\langle a,b,c\mid a^2b^2c^2\rangle$?
janmarqz
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Which small groups appear in multiple classical families like $C_n$, $D_n$, $S_n$, or $A_n$?

On page 90 of Visual Group Theory by Nathan Carter, there's an exercise (Exercise 5.20) that points out an interesting phenomenon: some small groups belong to more than one of the classical group families — cyclic $C_n$, dihedral $D_n$, symmetric…
F. A. Mala
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