The $\gamma$-Signless Laplacian Adjacency Matrix of Mixed Graphs

TL;DR

This paper introduces a new spectral matrix for mixed graphs, generalizing existing concepts, and studies its properties, including conditions for singularity and bounds related to eigenvalues.

Contribution

It defines a $oldsymbol{ ext{signless Laplacian adjacency matrix}}$ for mixed graphs and explores its spectral properties, extending previous adjacency matrix frameworks.

Findings

Characterizes when the matrix is singular.

Provides bounds on arcs and digons based on eigenvalues.

Analyzes spectral properties of the new matrix.

Abstract

The $\alpha$-Hermitian adjacency matrix $H_\alpha$ of a mixed graph $X$ has been recently introduced. It is a generalization of the adjacency matrix of unoriented graphs. In this paper, we consider a special case of the complex number $\alpha$. This enables us to define an incidence matrix of mixed graphs. Consequently, we define a generalization of line graphs as well as a generalization of the signless Laplacian adjacency matrix of graphs. We then study the spectral properties of the signless Laplacian adjacency matrix of a mixed graph. Lastly, we characterize when the signless Laplacian adjacency matrix of a mixed graph is singular and give lower and upper bounds of number of arcs and digons in terms of largest and lowest eigenvalue of the signless Laplacian adjacency matrix.