Groups all of whose undirected Cayley graphs are determined by their spectra

Abstract

Let $G$ be a finite group, and $S$ be a subset of $G\setminus\{1\}$ such that $S=S^{-1}$. Suppose that $Cay(G,S)$ is the Cayley graph on $G$ with respect to the set $S$ which is the graph whose vertex set is $G$ and two vertices $a,b\in G$ are adjacent if and only if $ab^{-1}\in S$. The adjacency spectrum $Spec(\Gamma)$ of a graph $\Gamma$ is the multiset of eigenvalues of its adjacency matrix. A graph $\Gamma$ is called "determined by its spectrum" (or for short DS) whenever if a graph $\Gamma'$ has the same spectrum as $\Gamma$, then $\Gamma \cong \Gamma'$. We say that the group $G$ is DS (Cay-DS, respectively) whenever if $\Gamma$ is a Cayley graph over $G$ and $Spec(\Gamma)=Spec(\Gamma')$ for some graph (Cayley graph, respectively) $\Gamma'$, then $\Gamma \cong \Gamma'$. In this paper, we study finite DS groups and finite Cay-DS groups. In particular we prove that all finite DS groups are solvable and all Sylow $p$-subgroups of a finite DS group is cyclic for all $p\geq 5$. We also give several infinite families of non Cay-DS solvable groups. In particular we prove that there exist two cospectral non-isomorphic $6$-regular Cayley graphs on the dihedral group of order $2p$ for any prime $p\geq 13$.