# Institute of Mathematics

## 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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References:
[1] R. Balaji, R. B. Bapat: Block distance matrices. Electron. J. Linear Algebra 16 (2007), 435-443. DOI 10.13001/1081-3810.1213 | MR 2365897 | Zbl 1148.15016
[2] R. B. Bapat: 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
[3] R. Bapat, S. J. Kirkland, M. Neumann: 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
[4] M. Fiedler: 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
[5] M. Fiedler, T. L. Markham: 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

Affiliations:   Balaji Ramamurthy, Department of Mathematics, IIT Madras, Tamil Nadu, Chennai, 600036, India, e-mail: balaji5@iitm.ac.in; Ravindra Bhalchandra Bapat, Theoretical Statistics and Mathematics Unit, Indian Statistical Institute, 7, S.J.S. Sansanwal Marg, New Delhi, 110016, India, e-mail: rbb@isid.ac.in; Shivani Goel, Department of Mathematics, IIT Madras, Tamil Nadu, Chennai, 600036, India, e-mail: shivani.goel.maths@gmail.com

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