Hermitian adjacency matrices of mixed multigraphs

TL;DR

This paper introduces a Hermitian matrix representation for mixed multigraphs, explores cospectral operations, and characterizes switching equivalence, providing bounds on cospectral classes sharing the same underlying graph.

Contribution

It presents a novel Hermitian matrix model for mixed multigraphs and characterizes their switching equivalence, advancing spectral graph theory for such structures.

Findings

Introduces a new Hermitian matrix representation for mixed multigraphs.

Provides a characterization of switching equivalent mixed multigraphs.

Establishes an upper bound on the number of cospectral classes with the same underlying graph.

Abstract

A mixed multigraph is obtained from an undirected multigraph by orienting a subset of its edges. In this paper, we study a new Hermitian matrix representation of mixed multigraphs, give an introduction to cospectral operations on mixed multigraphs, and characterize switching equivalent mixed multigraphs in terms of fundamental cycle basis. As an application, an upper bound of cospectral classes of mixed multigraphs with the same underlying graph is obtained.