Spectra of Cayley graphs

TL;DR

This paper establishes conditions under which Cayley graphs are integral, proving that certain normal subsets lead to graphs with integer eigenvalues, and applies this to specific symmetric groups.

Contribution

The paper provides new criteria for the integrality of Cayley graphs based on properties of the generating subset, including normality and element powers.

Findings

Cayley graphs with normal, power-closed subsets are integral.

Normal sets of involutions produce integral Cayley graphs.

Specific example with symmetric group A_n confirms the criteria.

Abstract

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".