Integral Cayley graphs over a finite symmetric algebra

A graph is called integral if its eigenvalues are integers. In this article, we provide 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 finite Hecke characters.