Integral Cayley graphs and groups

TL;DR

This paper classifies all finite groups with Cayley graphs that have only integer eigenvalues, solving two open problems and fully determining Cayley integral groups.

Contribution

It provides a complete classification of finite groups with Cayley graphs having integer eigenvalues and identifies all Cayley integral groups, advancing understanding in algebraic graph theory.

Findings

All finite groups with non-trivial Cayley graphs with integer eigenvalues are classified.

Complete determination of Cayley integral groups.

Resolution of two open problems in the classification of Cayley graphs.

Abstract

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.