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Distance Matrices Perturbed by Laplacians


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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  1. R. Balaji, R. B. Bapat: Block distance matrices. Electron. J. Linear Algebra 16 (2007), 435–443.

    MathSciNet  MATH  Google Scholar 

  2. R. B. Bapat: Determinant of the distance matrix of a tree with matrix weights. Linear Algebra Appl. 416 (2006), 2–7.

    Article  MathSciNet  Google Scholar 

  3. R. Bapat, S. J. Kirkland, M. Neumann: On distance matrices and Laplacians. Linear Algebra Appl. 401 (2005), 193–209.

    Article  MathSciNet  Google Scholar 

  4. M. Fiedler: Matrices and Graphs in Geometry. Encyclopedia of Mathematics and Its Applications 139. Cambridge University Press, Cambridge, 2011.

    Book  Google Scholar 

  5. M. Fiedler, T. L. Markham: Completing a matrix when certain entries of its inverse are specified. Linear Algebra Appl. 74 (1986), 225–237.

    Article  MathSciNet  Google Scholar 

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

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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).

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  • tree
  • Laplacian matrix
  • inertia
  • Haynsworth formula

MSC 2020

  • 05C50
  • 15B48