Let $G$ be a group and $S\subseteq G$ its subset such that $S=S^{ -1}$, where $S^{ -1}=\{ s^{ -1}\mid s\in S\}$. Then { \it the Cayley graph ${ \rm Cay}(G,S)$} is an undirected graph $\Gamma$ with the vertex set $V(\Gamma)=G$ and the edge set $E(\Gamma)=\{ (g,gs)\mid g\in G, s\in S\}$. A graph $\Gamma$ is said to be { \it integral} if every eigenvalue of the adjacency matrix of $\Gamma$ is integer. In the paper, we prove the following theorem: { \it if a subset $S=S^{ -1}$ of $G$ is normal and $s\in S\Rightarrow s^k\in S$ for every $k\in \mathbb{ Z}$ such that $(k,|s|)=1$, then ${ \rm Cay}(G,S)$ is integral.} In particular, { \it if $S\subseteq G$ is a normal set of involutions, then ${ \rm Cay}(G,S)$ is integral.} We also use the theorem to prove that { \it if $G=A_n$ and $S=\{ (12i)^{ \pm1}\mid i=3,\dots,n\}$, then ${ \rm Cay}(G,S)$ is integral.} Thus, we give positive solutions for both problems 19.50(a) and 19.50(b) in "Kourovka Notebook".
Categories: math.GR math.CO math.RT