More on minors of Hermitian (quasi-)Laplacian matrix of the second kind for mixed graphs

TL;DR

This paper investigates the properties of minors of Hermitian (quasi-)Laplacian matrices of the second kind for mixed graphs, providing explicit formulas and conditions related to spanning trees.

Contribution

It introduces new incidence matrices of the second kind for mixed graphs and characterizes minors of Hermitian (quasi-)Laplacian matrices, including conditions for cofactors.

Findings

Explicit expressions for minors of Hermitian (quasi-)Laplacian matrices.

Conditions under which cofactors equal the number of spanning trees.

Characterization of determinants for rootless mixed trees and unicyclic graphs.

Abstract

A mixed graph $M_{G}$ is the graph obtained from an unoriented simple graph $G$ by giving directions to some edges of $G$, where $G$ is often called the underlying graph of $M_{G}$. In this paper, we introduce two classes of incidence matrices of the second kind of $M_{G}$, and discuss the determinants of these two matrices for rootless mixed trees and unicyclic mixed graphs. Applying these results, we characterize the explicit expressions of various minors for Hermitian (quasi-)Laplacian matrix of the second kind of $M_{G}$. Moreover, we give two sufficient conditions that the absolute values of all the cofactors of Hermitian (quasi-)Laplacian matrix of the second kind are equal to the number of spanning trees of the underlying graph $G$.