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arXiv · 2207.07214

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

Qi Xiong·Gui-Xian Tian·Shu-Yu Cui
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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 $G$.

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Qi Xiong, Gui-Xian Tian, Shu-Yu Cui. 2022-07-14. More on minors of Hermitian (quasi-)Laplacian matrix of the second kind for mixed graphs. https://arxiv.org/abs/2207.07214

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