On finite groups all of whose cubic Cayley graphs are integral

Abstract

For any positive integer $k$, let ${\mathcal{G}}_k$ denote the set of finite groups $G$ such that all Cayley graphs $\mathrm{Cay}(G,S)$ are integral whenever $|S|\le k$. Est$\acute{\mathrm{e}}$lyi and Kov$\acute{\mathrm{a}}$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 $k\ge 4$.

11 pages, accepted by Journal of Algebra and its Applications on June 2015