Hermitian adjacency spectrum and switching equivalence of mixed graphs

TL;DR

This paper characterizes when an undirected graph is cospectral with a mixed graph's Hermitian adjacency matrix, introduces a four-way switching operation, and classifies rank-2 mixed graphs based on their spectral properties.

Contribution

It provides a spectral characterization of mixed graphs related to switching operations and classifies rank-2 mixed graphs by their Hermitian spectrum.

Findings

Undirected graph $G$ is cospectral with a mixed graph $D$ iff $H=G$ and $D$ is obtained by a four-way switching.

All rank-2 mixed graphs are determined by their Hermitian spectrum.

Identifies families of mixed graphs uniquely determined by their Hermitian spectrum.

Abstract

It is shown that an undirected graph $G$ is cospectral with the Hermitian adjacency matrix of a mixed graph $D$ obtained from a subgraph $H$ of $G$ by orienting some of its edges if and only if $H=G$ and $D$ is obtained from $G$ by a four-way switching operation; if $G$ is connected, this happens if and only if $\lambda_1(G)=\lambda_1(D)$. All mixed graphs of rank 2 are determined and this is used to classify which mixed graphs of rank 2 are cospectral with respect to their Hermitian adjacency matrix. Several families of mixed graphs are found that are determined by their Hermitian spectrum in the sense that they are cospectral precisely to those mixed graphs that are switching equivalent to them.