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More on Minors of Hermitian (Quasi-)Laplacian Matrix of the Second Kind for Mixed Graphs

COMPUTATIONAL & APPLIED MATHEMATICS(2024)

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摘要
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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关键词
Hermitian (quasi-)Laplacian matrix of the second kind,Matrix tree theorem,Mixed graphs,Spanning trees
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