Related papers: On Matrix Product Factorization of Cayley graphs

A digraph D is the pattern of a matrix M when D has an arc ij if and only if the ij-th entry of M is nonzero. Study the relationship between unitary matrices and their patterns is motivated by works in quantum chaology and quantum…

Let $G$ be a finite, undirected $d$-regular graph and $A(G)$ its normalized adjacency matrix, with eigenvalues $1 = \lambda_1(A)\geq \dots \ge \lambda_n \ge -1$. It is a classical fact that $\lambda_n = -1$ if and only if $G$ is bipartite.…

The unitary Cayley graph of $\mathbb{Z}/n\mathbb{Z}$, denoted $X_n$, is the graph on $\{0,\dots,n-1\}$ where vertices $a$ and $b$ are adjacent if and only if $\gcd(a-b,n) = 1$. We answer a question of Defant and Iyer by constructing a…

Recently in graph theory several authors have studied the spectrum of the Cayley graph of the symmetric group S_n generated by the transpositions (1, i) for 2 <= i <= n. Several conjectures were made and partial results were obtained. The…

We study graph products of groups from the viewpoint of measured group theory. We first establish a full measure equivalence classification of graph products of countably infinite groups over finite simple graphs with no transvection and no…

We say that a group G is a cube group if it is generated by a set S of involutions such that the corresponding Cayley graph Cay(G,S) is isomorphic to a cube. Equivalently, G is a cube group if it acts on a cube such that the action is…

The problem of classifying equivalence classes of presentations up to isomorphism of Cayley graphs is considered in this article in the case of dicyclic groups. The number of equivalence classes of presentations is uniformly bounded - it is…

Given a finite group $G$, the generating graph $\Gamma(G)$ of $G$ has as vertices the (nontrivial) elements of $G$ and two vertices are adjacent if and only if they are distinct and generate $G$ as group elements. In this paper we…

It has long been known that a vertex-transitive graph $\Gamma$ is isomorphic to a double coset graph $\text{Cos}(G,H,S)$ of a transitive group $G\le\text{Aut}(\Gamma)$, a vertex stabilizer $H\le G$, and some subset $S\subseteq G$. We show…

Let $G$ be a finite abelian group, let $E$ be a subset of $G$, and form the Cayley (directed) graph of $G$ with connecting set $E$. We explain how, for various matrices associated to this graph, the spectrum can be used to give information…

Let $G$ be a group and $S$ an inverse closed subset of $G\setminus \{1\}$. By a Cayley graph $Cay(G,S)$ we mean the graph whose vertex set is the set of elements of $G$ and two vertices $x$ and $y$ are adjacent if $x^{-1}y\in S$. A group…

We show that doubling at some large scale in a Cayley graph implies uniform doubling at all subsequent scales. The proof is based on the structure theorem for approximate subgroups proved by Green, Tao and the first author. We also give a…

In this paper we study the Cayley graph $\mathrm{Cay}(S_n,T)$ of the symmetric group $S_n$ generated by a set of transpositions $T$. We show that for $n\geq 5$ the Cayley graph is normal. As a corollary, we show that its automorphism group…

Let $\Gamma(G,S)$ denote the Cayley graph of a group $G$ with respect to a set $S \subset G$. In this paper, we analyze the spectral properties of the Cayley graphs $\mathcal{T}_{m,n,k} = \Gamma(\mathbb{Z}_m \ltimes_k \mathbb{Z}_n, \{(\pm…

Let $G$ be a finite non-abelian simple group, $C$ a non-identity conjugacy class of $G$, and $\Gamma_C$ the Cayley graph of $G$ based on $C \cup C^{-1}$. Our main result shows that in any such graph, there is an involution at bounded…

Given a finite group $G$ and a subset $S\subseteq G,$ the bi-Cayley graph $\bcay(G,S)$ is the graph whose vertex set is $G \times \{0,1\}$ and edge set is $\{\{(x,0),(s x,1)\} : x \in G, s\in S \}$. A bi-Cayley graph $\bcay(G,S)$ is called…

Let $\mathbb{U}$ be a Banach Lie group and $S\subseteq \mathbb{U}$ an ad-bounded subset thereof, in the sense that there is a uniform bound on the adjoint operators induced by elements of $S$ on the Lie algebra of $\mathbb{U}$. We prove…

A numerical semigroup $S$ is an additively-closed set of non-negative integers, and a factorization of an element $n$ of $S$ is an expression of $n$ as a sum of generators of $S$. It is known that for a given numerical semigroup $S$, the…

A finite group R is a CI-group if, whenever S and T are subsets of R with the Cayley graphs Cay(R,S) and Cay(R,T) isomorphic, there exists an automorphism x of R with S^x=T. The classification of CI-groups is an open problem in the theory…

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 $G$…