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

2022/07/14 by Qi Xiong, Xiong, Qi, Gui‐Xian Tian +3
Chemistry · Computer Science · Mathematics · #05C50 #15A18 #Combinatorics (math.CO) #FOS: Mathematics #Graph Labeling and Dimension Problems #Graph theory and applications #Synthesis and Properties of Aromatic Compounds

paper · pdf · doi:10.48550/arxiv.2207.07214

openalex publication_date 2022/07/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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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