HS-integral and Eisenstein integral mixed circulant graphs

TL;DR

This paper characterizes when mixed circulant graphs are HS-integral and Eisenstein integral, showing their equivalence and providing formulas for eigenvalues using generalized Möbius functions.

Contribution

It provides a complete characterization of HS-integral and Eisenstein integral mixed circulant graphs and establishes their equivalence, with explicit eigenvalue formulas.

Findings

Characterization of the set S for HS-integral mixed circulant graphs

Proof that Eisenstein integral and HS-integral properties are equivalent for these graphs

Explicit eigenvalue formulas involving generalized Möbius functions

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. We characterize the set $S$ for which a mixed circulant graph $\text{Circ}(\mathbb{Z}_n, S)$ is HS-integral. We also show that a mixed circulant graph is Eisenstein integral if and only if it is HS-integral. Further, the eigenvalues and the HS-eigenvalues of some oriented circulant graphs are expressed in terms of generalized M$\ddot{\text{o}}$bius function.