Related papers: The second eigenvalue of some normal Cayley graphs…

We construct a 2-generated group $\Gamma $ such that its Cayley graph possesses finite connected subsets with arbitrarily big finite Heesch number.

Let $\Gamma$ be a Cayley graph, or a Cayley sum graph, or a twisted Cayley graph, or a twisted Cayley sum graph, or a vertex-transitive graph. Denote the degree of $\Gamma$ by $d$, its edge Cheeger constant by $\mathfrak{h}_\Gamma$, and its…

An automorphism of a graph is called quasi-semiregular if it fixes a unique vertex of the graph and its remaining cycles have the same length. This kind of symmetry of graphs was first investigated by Kutnar, Malni\v{c}, Mart\'{i}nez and…

Let $G$ be a finite transitive group on a set $\Omega$, let $\alpha\in \Omega$ and let $G_\alpha$ be the stabilizer of the point $\alpha$ in $G$. In this paper, we are interested in the proportion $$\frac{|\{\omega\in \Omega\mid \omega…

We study the Steklov problem on a subgraph with boundary $(\Omega,B)$ of a polynomial growth Cayley graph $\Gamma$. We prove that for each $k \in \mathbb{N}$, the $k^{\mbox{th}}$ eigenvalue tends to $0$ proportionally to…

We study regular graphs whose distance-$2$ graph or distance-$1$-or-$2$ graph is strongly regular. We provide a characterization of such graphs $\Gamma$ (among regular graphs with few distinct eigenvalues) in terms of the spectrum and the…

We classify all the $2$-arc-transitive strongly regular graphs, and use this classification to study the family of finite $(G,3)$-geodesic-transitive graphs of girth $4$ or $5$ for some group $G$ of automorphisms. For this application we…

A mixed graph is said to be integral if all the eigenvalues of its Hermitian adjacency matrix are integer. Let $\Gamma$ be an abelian group. The \textit{mixed Cayley graph} $Cay(\Gamma,S)$ is a mixed graph on the vertex set $\Gamma$ and…

Let us say that a Cayley graph $\Gamma$ of a group $G$ of order $n$ is a Cerny Cayley graph if every synchronizing automaton containing $\Gamma$ as a subgraph with the same vertex set admits a synchronizing word of length at most $(n-1)^2$.…

Given a finite transitive permutation group $G\leq \operatorname{Sym}(\Omega)$, with $|\Omega|\geq 2$, the derangement graph $\Gamma_G$ of $G$ is the Cayley graph $\operatorname{Cay}(G,\operatorname{Der}(G))$, where $\operatorname{Der}(G)$…

We present a finer quantitative version of an observation due to Breuillard, Green, Guralnick and Tao which tells that for finite non-bipartite Cayley graphs, once the nontrivial eigenvalues of their normalized adjacency matrices are…

Let $A$ be an abelian group and let $\iota$ be the automorphism of $A$ defined by $i:a\mapsto a^{-1}$. A Cayley graph $\Gamma=\mathrm{Cay}(A,S)$ is said to have an automorphism group \emph{as small as possible} if $\mathrm{Aut}(\Gamma)=…

A perfect code $C$ in a graph $\Gamma$ is an independent set of vertices of $\Gamma$ such that every vertex outside of $C$ is adjacent to a unique vertex in $C$, and a total perfect code $C$ in $\Gamma$ is a set of vertices of $\Gamma$ such…

Let $G$ be a finite group, and let $x \in G$. Define $[x^G] := \{ y \in G : \langle x^G \rangle = \langle y^G \rangle \}$, where $\langle x^G \rangle$ denotes the normal subgroup of $G$ generated by the conjugacy class of $x$. In this…

In this paper, we propose the following conjecture which generalizes a theorem proved by Huang [Hua19] in his recent breakthrough proof of the sensitivity conjecture. We conjecture that for any Cayley graph $X = \Gamma(G,S)$ on a group $G$…

We show that for certain integers $n$, the problem of whether or not a Cayley digraph $\Gamma$ of $\mathbb Z_n$ is also isomorphic to a Cayley digraph of some other abelian group $G$ of order $n$ reduces to the question of whether or not a…

We study some properties of the Cayley graph of the R.Thompson's group F in generators $x_0$, $x_1$. We show that the density of this graph, that is, the least upper bound of the average vertex degree of its finite subgraphs is at least 3.…

For a non-bipartite finite Cayley graph, we show the non-trivial eigenvalues of its normalized adjacency matrix lie in the interval $$\left[-1+\frac{ch_{out}^2}{d},1-\frac{Ch_{out}^2}{d}\right],$$ for some absolute constant $c$ and $C$,…

The Star graph $S_n$, $n\ge 3$, is the Cayley graph on the symmetric group $Sym_n$ generated by the set of transpositions $\{(12),(13),\ldots,(1n)\}$. In this work we study eigenfunctions of $S_n$ corresponding to the second largest…

A graph $\Gamma$ is called edge-regular whenever it is regular and for any two adjacent vertices, the number of their common neighbors is independent of the choice of vertices. A clique $C$ in $\Gamma$ is called regular whenever for any…