Distance Powers and Distance Matrices of Integral Cayley Graphs over Abelian Groups

It is shown that distance powers of an integral Cayley graph over an abelian group are again integral Cayley graphs over that group. Moreover, it is proved that distance matrices of integral Cayley graphs over abelian groups have integral spectrum

Integral Cayley graphs and groups

We solve two open problems regarding the classification of certain classes of Cayley graphs with integer eigenvalues. We first classify all finite groups that have a "non-trivial" Cayley graph with integer eigenvalues, thus solving a problem proposed by Abdollahi and Jazaeri. The notion of Cayley integral groups was introduced by Klotz and Sander. These are groups for which every Cayley graph has only integer eigenvalues. In the second part of the paper, all Cayley integral groups are determined.Comment: Submitted June 18 to SIAM J. Discrete Mat

Groups all of whose undirected Cayley graphs are integral

Let $G$ be a finite group, $S\subseteq G\setminus\{1\}$ be a set such that if $a\in S$, then $a^{-1}\in S$, where $1$ denotes the identity element of $G$. The undirected Cayley graph $Cay(G,S)$ of $G$ over the set $S$ is the graph whose vertex set is $G$ and two vertices $a$ and $b$ are adjacent whenever $ab^{-1}\in S$. The adjacency spectrum of a graph is the multiset of all eigenvalues of the adjacency matrix of the graph. A graph is called integral whenever all adjacency spectrum elements are integers. Following Klotz and Sander, we call a group $G$ Cayley integral whenever all undirected Cayley graphs over $G$ are integral. Finite abelian Cayley integral groups are classified by Klotz and Sander as finite abelian groups of exponent dividing $4$ or $6$. Klotz and Sander have proposed the determination of all non-abelian Cayley integral groups. In this paper we complete the classification of finite Cayley integral groups by proving that finite non-abelian Cayley integral groups are the symmetric group $S_{3}$ of degree $3$, $C_{3} \rtimes C_{4}$ and $Q_{8}\times C_{2}^{n}$ for some integer $n\geq 0$, where $Q_8$ is the quaternion group of order $8$.Comment: Title is change