On finite groups all of whose cubic Cayley graphs are integral

TL;DR

This paper characterizes finite groups whose all cubic Cayley graphs are integral and extends the classification to groups where all Cayley graphs with degree up to three are integral, building on previous work for higher degrees.

Contribution

The paper provides a complete characterization of finite groups with all cubic Cayley graphs integral and revisits the classification of groups with all Cayley graphs of degree up to three integral.

Findings

Finite groups with all cubic Cayley graphs integral are characterized.

The class yi and Kove1cs' classification for  is extended.

The classification of  is obtained as a consequence.

Abstract

For any positive integer $k$, let $\mathcal{G}_k$ denote the set of finite groups $G$ such that all Cayley graphs ${\rm Cay}(G,S)$ are integral whenever $|S|\le k$. Est${\rm \acute{e}}$lyi and Kov${\rm \acute{a}}$cs \cite{EK14} classified $\mathcal{G}_k$ for each $k\ge 4$. In this paper, we characterize the finite groups each of whose cubic Cayley graphs is integral. Moreover, the class $\mathcal{G}_3$ is characterized. As an application, the classification of $\mathcal{G}_k$ is obtained again, where $k\ge 4$.