Integral Cayley graphs over a finite symmetric algebra

TL;DR

This paper characterizes when Cayley graphs over finite symmetric algebras are integral, extending previous results and exploring number-theoretic constructions related to global fields and Hecke characters.

Contribution

It provides necessary and sufficient conditions for integrality of Cayley graphs over finite symmetric algebras, generalizing prior work on rings modulo n.

Findings

Characterization of integral Cayley graphs over finite symmetric algebras

Extension of known results from rings modulo n to broader algebraic structures

Connections to number theory and potential applications to Paley graphs with Hecke characters

Abstract

A graph is called integral if its eigenvalues are integers. In this article, we provide the necessary and sufficient conditions for a Cayley graph over a finite symmetric algebra $R$ to be integral. This generalizes the work of So who studies the case where $R$ is the ring of integers modulo $n.$ We also explain some number-theoretic constructions of finite symmetric algebras arising from global fields, which we hope could pave the way for future studies on Paley graphs associated with a finite Hecke character.