Distance matrices perturbed by Laplacians

Ramamurthy, Balaji ; Bapat, Ravindra Bhalchandra ; Goel, Shivani

Applications of Mathematics, Tome 65 (2020) no. 5, pp. 599-607

Résumé

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 $L_{ii}:=\sum _{i \neq j,j=1}^{n}W_{ij}^{-1}$. 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.

Détail

DOI : 10.21136/AM.2020.0362-19

Classification : 05C50, 15B48
Keywords: tree; Laplacian matrix; inertia; Haynsworth formula