Resistance matrices of graphs with matrix weights

The resistance matrix of a simple connected graph G is denoted by R, and is defined by R=(rij), where rij is the resistance distance between the vertices i and j of G. In this paper, we consider the resistance matrix of weighted graph with edge weights being positive definite matrices of same size....

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Bibliographic Details
Published in:Linear algebra and its applications 2019-06, Vol.571, p.41-57
Main Authors: Atik, Fouzul, Bapat, R.B., Rajesh Kannan, M.
Format: Article
Language:English
Subjects:
Apexes
Eigenvalues
Inertia
Inverse
Laplacian matrix
Linear algebra
Mathematical analysis
Matrix methods
Matrix weighted graph
Moore–Penrose inverse
Resistance matrix
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Summary:The resistance matrix of a simple connected graph G is denoted by R, and is defined by R=(rij), where rij is the resistance distance between the vertices i and j of G. In this paper, we consider the resistance matrix of weighted graph with edge weights being positive definite matrices of same size. We derive a formula for the determinant and the inverse of the resistance matrix. Then, we establish an interlacing inequality for the eigenvalues of resistance and Laplacian matrices of tree. Using this interlacing inequality, we obtain the inertia of the resistance matrix of tree.
ISSN:0024-3795
1873-1856
DOI:10.1016/j.laa.2019.02.011