On Hermitian Adjacency Matrices for Mixed Graphs

TL;DR

This paper investigates the spectral properties of mixed graphs using Hermitian adjacency matrices, introduces a new matrix variant, and establishes bounds on spectral radius and conditions for cospectrality.

Contribution

It extends existing results on Hermitian adjacency matrices for mixed graphs, introduces a new matrix with roots of unity, and characterizes spectral bounds and cospectrality conditions.

Findings

Characterization of cospectrality conditions for mixed and underlying graphs

Determination of a sharp upper bound on spectral radius

Identification of extremal graphs achieving the bound

Abstract

We study the spectra of mixed graphs about its Hermitian adjacency matrix of the second kind (i.e. N-matrix) introduced by Mohar [1]. We extend some results and define one new Hermitian adjacency matrix, and the entry corresponding to an arc from $u$ to $v$ is equal to the $k$-th( or the third) root of unity, i.e. ${\omega} = cos(2{\pi}/k) + \textbf{i} \ sin(2{\pi}/k), k {\geq} 3$; the entry corresponding to an undirected edge is equal to 1, and 0 otherwise. In this paper, we characterize the cospectrality conditions for a mixed graph and its underlying graph. In section 4, we determine a sharp upper bound on the spectral radius of mixed graphs, and provide the corresponding extremal graphs.