HS-integral and Eisenstein integral mixed Cayley graphs over abelian groups

TL;DR

This paper characterizes when mixed Cayley graphs over abelian groups have integer or Eisenstein integer eigenvalues, establishing a connection between HS-integrality and Eisenstein integrality.

Contribution

It provides a complete characterization of HS-integral mixed Cayley graphs over abelian groups and proves their equivalence to Eisenstein integral graphs.

Findings

Characterization of the set S for HS-integral mixed Cayley graphs.

Proof that Eisenstein integrality is equivalent to HS-integrality for these graphs.

Theoretical framework linking eigenvalue properties to group structure.

Abstract

A mixed graph is called \emph{second kind hermitian integral}(or \emph{HS-integral}) if the eigenvalues of its Hermitian-adjacency matrix of second kind are integers. A mixed graph is called \emph{Eisenstein integral} if the eigenvalues of its (0, 1)-adjacency matrix are Eisenstein integers. Let $\Gamma$ be an abelian group. We characterize the set $S$ for which a mixed Cayley graph $\text{Cay}(\Gamma, S)$ is HS-integral. We also show that a mixed Cayley graph is Eisenstein integral if and only if it is HS-integral.