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

16 pages,7 figures