Related papers: A formula for eigenvalues of integral Cayley graph…

The sum of the absolute values of the eigenvalues of a graph is called the energy of the graph. We study the problem of finding graphs with extremal energy within specified classes of graphs. We develop tools for treating such problems and…

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…

Binomial Cayley graphs are obtained by considering the binomial coefficient of the weight function of a given Cayley graph and a natural number. We introduce these objects and study two families: one associated with symmetric groups and the…

A Cayley graph of a group $H$ is a finite simple graph $\Gamma$ such that its automorphism group ${\rm Aut}(\Gamma)$ contains a subgroup isomorphic to $H$ acting regularly on $V(\Gamma)$, while a Haar graph of $H$ is a finite simple…

A perfect code in a graph $\Gamma = (V, E)$ is a subset $C$ of $V$ such that no two vertices in $C$ are adjacent and every vertex in $V \setminus C$ is adjacent to exactly one vertex in $C$. A total perfect code in $\Gamma$ is a subset $C$…

The unitary addition Cayley graph $G_n$, $n\in Z^+$ is the graph whose vertex set is $Z_n$, the ring of integers modulo $n$ and two vertices $u$ and $v$ are adjacent if and only if $u + v \in \cup_n$ where $\cup_n$ is the set of all units…

Let $\Gamma$ be a simple connect graph on a finite vertex set $V$ and let $A$ be its adjacency matrix. Then $\Gamma$ is said to be \textit{singular} if and only if $0$ is an eigenvalue of $A.$ The \textit{nullity (singularity)} of $\Gamma,$…

A number of authors have studied the question of when a graph can be represented as a Cayley graph on more than one nonisomorphic group. The work to date has focussed on a few special situations: when the groups are $p$-groups; when the…

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…

A finite group $G$ is said to be Cayley integral if every undirected Cayley graph $\operatorname{Cay}(G,S)$ on $G$ is integral. In this paper, we introduce three natural extensions of this concept; namely as: Cayley colour integral,…

Given a countable abelian group $A$, we construct a row finite directed graph $\Gamma(A)$ such that the $K_{0}$-group of the graph $\textrm{C}^{\ast}$-algebra $\textrm{C}^{\ast}(\Gamma(A))$ is canonically isomorphic to $A$. Moreover, each…

Cayley graphs have a number of useful features: the ability to graphically represent finitely generated group elements and their relations; to name all vertices relative to a point; and the fact that they have a well-defined notion of…

Let $G$ be a finite abelian group of order $n$. For any subset $B$ of $G$ with $B=-B$, the Cayley graph $G_B$ is a graph on vertex set $G$ in which $ij$ is an edge if and only if $i-j\in B.$ It was shown by Ben Green that when $G$ is a…

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…

Isomorphic factorizations of complete graphs originate from the seminal work of Frank Harary and collaborators, who initiated the systematic study of decompositions of complete graphs into pairwise isomorphic spanning subgraphs. In this…

Dual Seidel switching is a graph operation introduced by W.~Haemers in 1984. This operation can change the graph, however it does not change its bipartite double, and because of this, the operation leaves the squares of the eigenvalues…

A graph is called a GRR if its automorphism group acts regularly on its vertex-set. Such a graph is necessarily a Cayley graph. Godsil has shown that there are only two infinite families of finite groups that do not admit GRRs : abelian…

We give a sufficient and necessary condition for a Praeger-Xu graph to be a Cayley graph.

The Cayley sum graph $\Gamma_S$ of a set $S \subseteq \mathbb{Z}_n$ is defined on the vertex set $\mathbb{Z}_n$, with an edge between distinct $x, y \in \mathbb{Z}_n$ if $x + y \in S$. Campos, Dahia, and Marciano have recently shown that if…

A finite simple graph is called a bi-Cayley graph over a group $H$ if it has a semiregular automorphism group, isomorphic to $H,$ which has two orbits on the vertex set. Cubic vertex-transitive bi-Cayley graphs over abelian groups have been…