HS-integral and Eisenstein integral normal mixed Cayley graphs

TL;DR

This paper characterizes when normal mixed Cayley graphs are HS-integral and Eisenstein integral, establishing their equivalence and generalizing previous results to broader classes of groups.

Contribution

It provides a complete characterization of the sets S for HS-integrality and proves the equivalence with Eisenstein integrality in normal mixed Cayley graphs.

Findings

Characterization of S for HS-integral Cayley graphs.

Equivalence of HS-integral and Eisenstein integral properties.

Generalization of previous results to non-abelian groups.

Abstract

A mixed graph is said to be HS-\emph{integral} if the eigenvalues of its Hermitian-adjacency matrix of the second kind are integers. A mixed graph is called \emph{Eisenstein integral} if the eigenvalues of its (0, 1)-adjacency matrix are Eisenstein integers. We characterize the set $S$ for which the normal mixed Cayley graph $\text{Cay}(\Gamma, S)$ is HS-integral for any finite group $\Gamma$. We further show that a normal mixed Cayley graph is HS-integral if and only if it is Eisenstein integral. This paper generalizes the results of [M. Kadyan, B. Bhattacharjya. HS-integral and Eisenstein integral mixed Cayley graphs over abelian groups. Linear Algebra Appl. 645:68-90, 2022].