On the inverse and Moore–Penrose inverse of resistance matrix of graphs with more general matrix weights

Article Type

Research Article

Publication Title

Journal of Applied Mathematics and Computing

Abstract

Let G represent a simple connected graph with a vertex set denoted as { 1 , 2 , 3 , … , n} . The Laplacian matrix of G is denoted as L. The resistance distance of two vertices i and j is given by the expression: rij=lii†+ljj†-2lij†,i,j=1,2,3,⋯,n , where L†=(lij†)n×n represents the Moore–Penrose inverse of matrix L. The resistance matrix of the graph G is defined by R=(rij)n×n . Prior research in the literature has extensively explored determinants and inverses of the resistance matrix. This article extends the concept of a resistance matrix to incorporate matrix weights, particularly when dealing with symmetric matrix edge weights. It is demonstrated that the resistance matrix may lose its non-singularity property under this setup. The conditions for singularity of the resistance matrix R are established as necessary and sufficient. In cases where the resistance matrix becomes singular, the potential ranks of matrix R are characterized. An explicit equation is formulated to calculate the inverse and determinant of the resistance matrix in cases where it is not singular. Furthermore, for a specific scenario, the Moore–Penrose inverse of a singular resistance matrix is provided.

First Page

4805

Last Page

4820

DOI

https://10.1007/s12190-023-01945-w

Publication Date

12-1-2023

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