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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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 ...
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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,...
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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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Infinitely many finitely generated groups having the same Cayley graph
Is there an unlabeled locally-finite graph which is a Cayley graph of an infinitely many non-isomorphic groups with respect to suitably chosen generating sets?
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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 ...
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16
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1
answer
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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 ...
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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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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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2
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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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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 ...
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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 $G$...
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1
answer
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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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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 ...
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1
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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 $k(g,h) = \...
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$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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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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1
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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 ...
5
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1
answer
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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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(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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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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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 ...
- 1,841
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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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4
votes
2
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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)...
- 1,841
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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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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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1
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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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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 ...
- 1,841
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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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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),\\...
- 1,841
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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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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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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=...
- 1,841
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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 (...
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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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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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3
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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 ...
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2
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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 ...
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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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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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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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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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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 ...
- 1,841
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vote
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 ...
- 1,841
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vote
0
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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 $2k$ ...
- 1,841
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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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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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0
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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 ...
- 1,841
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0
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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 ...
- 1,841
0
votes
2
answers
134
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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 ...
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