On distance and Laplacian matrices of trees with matrix weights
Linear and Multilinear Algebra
The distance matrix of a simple connected graph G is (Formula presented.), where (Formula presented.) is the distance between the vertices i and j in G. We consider a weighted tree T on n vertices with edge weights are square matrices of the same size. The distance (Formula presented.) between the vertices i and j is the sum of the weight matrices of the edges in the unique path from i to j. In this article, we establish a characterization for the trees in terms of the rank of (matrix) weighted Laplacian matrix associated with it. We present a necessary and sufficient condition for the distance matrix D, with matrix weights, to be invertible and the formula for the inverse of D, if it exists. Then we study some of the properties of the distance matrices of matrix weighted trees in connection with the Laplacian matrices, incidence matrices, and g-inverses. Finally, we derive an interlacing inequality for the eigenvalues of distance and Laplacian matrices for the case of positive definite matrix weights.
Atik, Fouzul; Kannan, M. Rajesh; and Bapat, Ravindra B., "On distance and Laplacian matrices of trees with matrix weights" (2021). Journal Articles. 2271.
Open Access, Green