1. Hermitian adjacency matrix of the second kind for mixed graphs
- Author
-
Shuchao Li and Yuantian Yu
- Subjects
FOS: Mathematics ,Mathematics - Combinatorics ,Discrete Mathematics and Combinatorics ,Combinatorics (math.CO) ,05C35, 05C12 ,Theoretical Computer Science - 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 $\alpha\in\left\{\sqrt{2},\,\sqrt{3},\,2\right\}$., Comment: 31pages,15figures
- Published
- 2022