Integral Cayley graphs over a finite symmetric algebra
Tung T. Nguyen, Nguyen Duy Tân
TL;DR
This work classifies when Cayley graphs $\Gamma(R,S)$ over finite symmetric $\mathbb{Z}/n$-algebras $R$ are integral, i.e., have all eigenvalues in $\mathbb{Z}$. Using a non-degenerate linear functional and the associated $R$-DFT matrix $A_R$, the authors show that $S=-S$ with stability under $(\mathbb{Z}/n)^{\times}$ guarantees integrality, generalizing So's theorem from $R=\mathbb{Z}/n$; conversely, integrality implies $S$ is a union of $(\mathbb{Z}/n)^{\times}$-orbits. The paper provides broad constructions of symmetric $\mathbb{Z}/n$-algebras (including group algebras and finite quotients of rings of integers in global fields) and demonstrates the applicability of the criterion to generalized Paley graphs $P_{\chi}$, tying spectral graph theory to algebraic number theory and Hecke characters. This framework enables systematic generation of integral Cayley graphs over a wide class of finite rings and highlights avenues for further study of Paley-type graphs in arithmetic settings.
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.