Hermitian adjacency matrix of the second kind for mixed graphs

TL;DR

This paper studies the spectral properties of mixed graphs using a specific Hermitian adjacency matrix, providing conditions for cospectrality, bounds on spectral radius, and classifications of graphs based on matrix rank and eigenvalue ranges.

Contribution

It introduces new spectral characterizations and classifications of mixed graphs via the $N$-matrix, including extremal graphs, switching operations, and eigenvalue range analyses.

Findings

Characterization of cospectral mixed graphs with the same underlying graph.

Sharp upper bound on the spectral radius and extremal graphs.

Complete classification of graphs with rank 2 and 3, and eigenvalues in specified ranges.

Abstract

This contribution gives an extensive study on spectra of mixed graphs via its Hermitian adjacency matrix of the second kind { ($N$-matrix for short)} introduced by Mohar \cite{0001}. This matrix is indexed by the vertices of the mixed graph, and the entry corresponding to an arc from $u$ to $v$ is equal to the sixth root of unity $\omega=\frac{1+{\bf i}\sqrt{3}}{2}$ (and its symmetric entry is $\bar{\omega}=\frac{1-{\bf i}\sqrt{3}}{2}$); the entry corresponding to an undirected edge is equal to 1, and 0 otherwise. The main results of this paper include the following: {equivalent} conditions for a mixed graph that shares the same spectrum of its $N$-matrix with its underlying graph are given. A sharp upper bound on the spectral radius is established and the corresponding extremal mixed graphs are identified. Operations which are called two-way and three-way switchings are discussed--they give rise to some cospectral mixed graphs. We extract all the mixed graphs whose rank of its $N$-matrix is $2$ (resp. 3). Furthermore, we show that {if $M_G$ is a connected mixed graph with rank $2,$ then $M_G$ is switching equivalent to each connected mixed graph to which it is cospectral}. However, this does not hold for some connected mixed graphs with rank $3$. We identify all mixed graphs whose eigenvalues of its $N$-matrix lie in the range $(-\alpha,\, \alpha)$ for $\alpha\in\left\{\sqrt{2},\,\sqrt{3},\,2\right\}$.