# Article

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Keywords:
tree; Laplacian matrix; inertia; Haynsworth formula
Summary:
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.
References:
 Balaji, R., Bapat, R. B.: Block distance matrices. Electron. J. Linear Algebra 16 (2007), 435-443. DOI 10.13001/1081-3810.1213 | MR 2365897 | Zbl 1148.15016
 Bapat, R. B.: Determinant of the distance matrix of a tree with matrix weights. Linear Algebra Appl. 416 (2006), 2-7. DOI 10.1016/j.laa.2005.02.022 | MR 2232916 | Zbl 1108.15006
 Bapat, R., Kirkland, S. J., Neumann, M.: On distance matrices and Laplacians. Linear Algebra Appl. 401 (2005), 193-209. DOI 10.1016/j.laa.2004.05.011 | MR 2133282 | Zbl 1064.05097
 Fiedler, M.: Matrices and Graphs in Geometry. Encyclopedia of Mathematics and Its Applications 139. Cambridge University Press, Cambridge (2011). DOI 10.1017/CBO9780511973611 | MR 2761077 | Zbl 1225.51017
 Fiedler, M., Markham, T. L.: Completing a matrix when certain entries of its inverse are specified. Linear Algebra Appl. 74 (1986), 225-237. DOI 10.1016/0024-3795(86)90125-4 | MR 0822149 | Zbl 0592.15002