Integral Cayley multigraphs over Abelian and Hamiltonian groups

TL;DR

This paper characterizes when Cayley multigraphs over Abelian and Hamiltonian groups are integral, linking eigenvalues to the structure of generating sets and providing new character theoretic proofs.

Contribution

It offers a new characterization of integral Cayley multigraphs over Abelian and Hamiltonian groups using character sums and subgroup structures.

Findings

Integral Cayley multigraphs over Abelian groups are characterized by generating sets in the integral cone.

A necessary and sufficient condition for Hamiltonian groups is established based on character sums.

Provides an alternative proof of a classical theorem using character theory.

Abstract

It is shown that a Cayley multigraph over a group $G$ with generating multiset $S$ is integral (i.e., all of its eigenvalues are integers) if $S$ lies in the integral cone over the boolean algebra generated by the normal subgroups of $G$. The converse holds in the case when $G$ is abelian. This in particular gives an alternative, character theoretic proof of a theorem of Bridges and Mena (1982). We extend this result to provide a necessary and sufficient condition for a Cayley multigraph over a Hamiltonian group to be integral, in terms of character sums and the structure of the generating set.