# Distance Matrices Perturbed by Laplacians

## 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 Dij denote the sum of all the weights lying in the path connecting the vertices i and j of T. We now say that Dij is the distance between i and j. Define D ≔ [Dij], where Dii is the s × s null matrix and for ij, Dij 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 Lij ≔ − W−1ij , where Wij is the weight of the edge (i, j). Define $${L_{ii}}: = \sum\limits_{i \ne j,j = 1}^n {W_{ij}^{ - 1}}.$$ The Laplacian of G is now the ns × ns block matrix L ≔ [Lij]. In this paper, we first note that D−1L is always nonsingular and then we prove that D and its perturbation (D−1L)−1 have many interesting properties in common.

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## References

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## Author information

Authors

### Corresponding authors

Correspondence to Balaji Ramamurthy, Ravindra Bhalchandra Bapat or Shivani Goel.

The first author is supported by Department of science and Technology, India under the project MATRICS (MTR/2017/000342).

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Ramamurthy, B., Bapat, R.B. & Goel, S. Distance Matrices Perturbed by Laplacians. Appl Math 65, 599–607 (2020). https://doi.org/10.21136/AM.2020.0362-19

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• DOI: https://doi.org/10.21136/AM.2020.0362-19

### Keywords

• tree
• Laplacian matrix
• inertia
• Haynsworth formula

• 05C50
• 15B48