# 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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### How good is approximation of distance function on the Cayley graph by "Fourier" basis coming from the irreducible representations?

Consider finite group $G$ , symmetric set of its elements $S$, construct a Cayley graph.
Consider $d(g)$ - word metric or distance on the Cayley graph from identity to $g$.
As any function on a group ...

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### How to choose N policemen positions to catch a drunk driver in the most effective way (on a Cayley graph of a finite group)?

Consider a Cayley graph of some big finite group. Consider random walk on such a graph - think of it as drunk driver. Fix some number $N$ which is much smaller than group size.
Question 1: How to ...

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### Necessary and sufficient conditions for the Cayley graph to be bipartite

Let $G$ be a finite group with identity $1$. If $S$ be an inverse closed generated subset of $G$, then $S$ is called a Cayley subset of $G$.The Cayley graph $\Gamma=\operatorname{Cay}(G, S)$ is a ...

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### Short path problem on Cayley graphs as language translation task (from "Permutlandski" to "Cayleylandski"(s) :). Reference/suggestion request

Context: Algorithms to find short paths on Cayley graphs of (finite) groups are of some interest - see below.
There can be several approaches to that task. One of ideas coming to my mind - in some ...

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### Does the permutohedron satisfy any minimal distortion property for graph metric vs Euclidean distance?

We can look on the permutohedron as a kind of "embedding" of the Cayley graph of $S_n$ to the Euclidean space. (That Cayley graph is constructed by the standard generators, i.e. ...

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### What Cayley graphs arise as nodes+edges from "nice" polytopes and when are these polytopes convex?

The Permutohedron is a remarkable convex polytope in $R^n$, such that its nodes are indexed by permutations and edges correspond to the Cayley graph of $S_n$ with respect to the standard generators, i....

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### Growth of spheres in FINITE nilpotent groups - Gaussian approximation (central limit theorem)?

Standard setup. Consider a group and choose generators. Word-metric (or in the other words - distance on the Cayley graph of the group+generators) - converts a group into a metric space, which is ...

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### Polynomials of growth for finite Heisenberg groups

Take a standard finite Heisenberg group with two standard generators and let's consider its growth polynomial - the polynomial which coefficients are equal to the sphere sizes.
For example for $H_3(Z/...

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### Commutative Schur ring over non-abelian group

Is there a commutative (or even symmetric) Schur ring $S\subset\mathbb{C}G$ over a non-abelian group $G$, which is not isomorphic (preserving both the products) to a Schur ring $S'\subset\mathbb{C}G'$ ...

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### Open problems which might benefit from computational experiments

Question: I wonder what are the open problems , where computational experiments might me helpful? (Setting some bounds, excluding some cases, shaping some expectations ).
Grant program: The context of ...

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### Diameter of the "Masterball-puzzle" permutation groups by a kind of Cartier-Foata enumeration?

There is an wonderful blog post by Jordan S. Ellenberg SHOULD YOU BE SURPRISED BY THE DIAMETER OF THE NXNXN RUBIK’S GROUP?. Which explains how one can come to $N^2log(N)$ estimate of the diameter of ...

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### Distance distribution for Cayley graphs of the fintie Heisenberg groups H3(Z/nZ) approaches Gaussian for large "n"?

I wonder several questions about Cayley graphs of finite Heisenberg groups H3(Z/nZ).
Question 1: do we know the diameter dependence on "n", at least for the standard choice of generators ? ...

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### Distances on spheres in Cayley graphs of non-amenable groups

Let $G$ be a non-amenable group (or perhaps more generally, a group with exponential growth). For any $\epsilon>0$, define the shell of radius r, $S_\epsilon(r)$, as the set of points that lie at a ...

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### Edge coloring of a graph on alternating groups

Let $G$ be the Cayley graph on the alternating group $A_n\,n\ge4$ with generating set $$S=\begin{cases}\{(1,2,3),(1,3,2),\\(1,2,\ldots,n),(1,n,n-1,\ldots,2)\}, &n\ \text{odd}\\ \{(1,2,3),(1,3,2),\\...

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### Does every infinite, connected, locally finite, vertex-transitive graph have a leafless spanning tree?

My question is
Let $G$ be an infinite, connected, locally finite, vertex-transitive graph. Must
$G$ have the following substructures?
i) a leafless spanning
tree;
ii) a spanning forest consisting ...

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### (How) does the spectral gap of the $n\times n$ Rubik's cube close with $n$?

Consider the spectrum of the adjacency matrix $A$ of the Cayley graph of the standard, 3x3x3 Rubik's cube generated with the usual quarter-turn and half-turn twists of each face (the Singmaster ...

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### Does the visual boundary of any one-ended Cayley graph contain at least three points?

Let $\Gamma$ be a Cayley graph of a finitely generated group. We can define the visual boundary of $\Gamma$ with respect to some base vertex $b$, denoted $\partial \Gamma$, as the set of geodesic rays ...

3
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### Property $(T)$ for $\mathrm{SL}_2(\mathbb{Z}) \ltimes \mathbb{Z}^2$

(This is in part a request for references and in part a somewhat pedagogical question.)
I gave a course on expanders seven years ago, and I am giving a course on expanders again now. We will soon do ...

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### Bandwidth of finite groups

For a generating set $S$ of a group $G$ denote by $\mathrm{Cay}(G,S)$ the corresponding Cayley graph.
For a finite graph $A$ denote by $\beta(A)$ its bandwidth.
Question: Has the "group bandwidth&...

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### Groups of non-orientable genus 1 and 2

The non-orientable genus (aka crosscap-number) $\overline{\gamma}(G)$ of a finite group $G$ is the minimum non-orientable genus among all its connected Cayley graphs (and $0$ if $G$ has a planar ...

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### Chromatic numbers of Cayley graphs induced by Hamming balls

The motivation for this question is to find, for a fixed odd $p$ and large $n$, sets $A\subset (\mathbb Z/p\mathbb Z)^n$ with $|A|> cp^n$ for some fixed $c$, where the difference set $A-A:=\{a-a': ...

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### Bruhat-Tits tree as Cayley graph of free group

$\DeclareMathOperator\BT{BT}\DeclareMathOperator\GL{GL}$Let $p > 2$ be a prime and $n = \frac{p + 1}{2}$. We can identify the vertices of Bruhat-Tits tree $\BT(\mathbb Q_p)$ with the elements in ...

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### Hamiltonian cycles in Cayley graph on alternating group

Let $G=\operatorname{Cay}(A_n,S)$ be the Cayley graph on the Alternating group $A_n\quad n\ge4$ with generating set $S=\{(1,2,3),(1,2,4),\ldots,(1,4,2),(1,3,2)\}$. One Hamiltonian cycle in $G$ for $n=...

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### What about a Cayley n-complex for n>2?

Let $G$ be a finitely presented group. The Cayley graph of the finite generating set is a $1$-complex where the $0$-cells are the elements of $G$ and the $1$-cells are given by the generators (...

3
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1
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### Explicit formula for embedding Cayley graph of free group into hyperbolic space

The problem is to embed Cayley graph of free group with $n\geq2$ generators (the same as Bethe lattice with coordination number $2n$) into any model of $\mathbb{H}^2$ (we have no model preference, the ...

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### When is a Schreier coset graph vertex transitive

When is a Schreier Coset graph on a group $G$ with subgroup $H$ and symmetric generating set $S$(without identity) vertex transitive?
It is well known that when $H$ is normal, the Schreier coset graph ...

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### Difference in chromatic number between Schreier coset graphs and Cayley graphs

Can the Schreier coset graphs can be seen as a subgraph of Cayley graph on the same groups(neglecting the loop edges) and, hence, have their chromatic numbers bounded by the chromatic numbers of the ...

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### Imposing reciprocity in the definition of vertex-transitivity

A simple, undirected graph is vertex-transitive if for any pair of vertices $x,y$, there exists an automorphism (adjacency-preserving self-bijection) $\phi$ such that $\phi(x)=y$.
What if, instead of ...

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1
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### Bound on chromatic number of graphs on any finite $p$-group

Is the chromatic number of a Cayley graph on $p$-groups with any generating set bounded by the chromatic number of the maximal induced circulant subgraph?
I think yes. Because for one, the main ...

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1
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### Extending the vertex coloring of circulant graph to graph on $p$-group

Let $G_1$ be a circulant graph of prime order $p$. This implies that $G_1$ is the Cayley graph on $\mathbb{Z}_p$ with some generating set $S_1$. I am interested in knowing the characterizations of the ...

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### Circulant graphs chromatically dominated by powers of cycles

Suppose we can color the vertices of powers of cycles $C_n^k$ using $c$ colors such that each of the color classes $c_i$ have $v_i$ number of vertices. Can we always color the circulant of degree $2k$ ...

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### halved and folded hypercube duality

Notation. Consider the group $\Gamma=\mathbb{Z}_2^n$. I will denote the group operation aditively and by $\epsilon_i=(0,\dots,0,1,0,\dots,0)$ I denote the canonical generators. Let's define also $\...

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### Is the function $k(g,h) = \frac{1}{1+\lvert gh^{-1}\rvert}$ positive definite?

Let $G$ be a finite group, $S \subset G$ a generating set, closed under taking inverses, and $\lvert\cdot\rvert$ the word length with respect to this set $S$.
Question. Is the function $k(g,h) = \...

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### Is the Petersen graph a "Cayley graph" of some more general group-like structure?

The Petersen graph is the smallest vertex-transitive graph which is not a Cayley graph. Is it the "Cayley graph" of some slightly more general group-like structure?

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### Example of family of Cayley graphs with Ramanujan behaviour on finite $p$-groups

This is a very general question: are there known examples of Ramanujan behaviour of Cayley graphs obtained from family of finite p-groups?
${\mathrm{\bf Adjacency~matrix:}}$ Given a graph ${\mathcal{G}...

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### Procedure to color the edges of a circulant graph

From the first theorem in this paper, it is clear that a cayley graph on abelian group for all generating sets of even order is class $1$, that is can be edge colored in exactly $\Delta$ colors. But, ...

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### how do I find eigenvalues of Cayley graph for one subset given a different subset

How do I find eigenvalues for the adjacency matrix of Cayley graph $X(S_n,S)$ where $S_n$ is the symmetric group of order $n$ and $S$ is the set of transpositions $(i,i+1)$, if the eigenvalues of the ...

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### Which line graphs of Cayley graphs are Cayley

When are the line graphs of Cayley graphs Cayley?
From this link we can know when the line graphs of complete graphs are cayley. But, my question pertains to the larger class of Cayley graphs. Are ...

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### Cayley graphs on $Z_{11}$ and $Z_p$

I want to find all cayley graphs on $Z_{11}$. I know how many connected cayley graphs exist but i want to find all of them, connected or not, to find their eigenvalues. I found some of them and a ...

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### Path-covering for vertex-transitive graphs

I have the following dummy problem:
Claim - There exists $N$ such that for $n > N$, if $G_n$ be a connected directed vertex-transitive graph with $n$ vertices, then there exists a set $S$ of paths ...

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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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1
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### 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 ...

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### 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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### Recognizing perfect Cayley graphs as tensor products

It is known (and can easily be seen) that a unitary Cayley graph on $n=\prod_ip_i$, ($p_i$ distinct primes) vertices with $n$ square-free can be recognized as the tensor product of the graphs $K_{p_i}$...

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### $C_4\times C_2 : C_2$: 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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### 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 ...

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### 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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### 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 ...

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### 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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### 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 ...