Applications of Mathematics, Vol. 65, No. 5, pp. 599-607, 2020

Distance matrices perturbed by Laplacians

Balaji Ramamurthy, Ravindra Bhalchandra Bapat, Shivani Goel

Received December 14, 2019.   Published online September 4, 2020.

Abstract:  Let $T$ be a tree with $n$ vertices. To each edge of $T$ we assign a weight which is a positive definite matrix of some fixed order, say, $s$. Let $D_{ij}$ denote the sum of all the weights lying in the path connecting the vertices $i$ and $j$ of $T$. We now say that $D_{ij}$ is the distance between $i$ and $j$. Define $D:=[D_{ij}]$, where $D_{ii}$ is the $s \times s$ null matrix and for $i \neq j$, $D_{ij}$ is the distance between $i$ and $j$. Let $G$ be an arbitrary connected weighted graph with $n$ vertices, where each weight is a positive definite matrix of order $s$. If $i$ and $j$ are adjacent, then define $L_{ij}:=-W_{ij}^{-1}$, where $W_{ij}$ is the weight of the edge $(i,j)$. Define $L_{ii}:=\sum_{i \neq j,j=1}^nW_{ij}^{-1}$. The Laplacian of $G$ is now the $ns \times ns$ block matrix $L:=[L_{ij}]$. In this paper, we first note that $D^{-1}-L$ is always nonsingular and then we prove that $D$ and its perturbation $(D^{-1}-L)^{-1}$ have many interesting properties in common.
Keywords:  tree; Laplacian matrix; inertia; Haynsworth formula
Classification MSC:  05C50, 15B48
DOI:  10.21136/AM.2020.0362-19

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Affiliations:   Balaji Ramamurthy, Department of Mathematics, IIT Madras, Tamil Nadu, Chennai, 600036, India, e-mail:; Ravindra Bhalchandra Bapat, Theoretical Statistics and Mathematics Unit, Indian Statistical Institute, 7, S.J.S. Sansanwal Marg, New Delhi, 110016, India, e-mail:; Shivani Goel, Department of Mathematics, IIT Madras, Tamil Nadu, Chennai, 600036, India, e-mail:

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