The distance matrix of a bidirected tree

Bapat, R.B.; Lal, A. K.; Pati, S.

doi:10.13001/1081-3810.1308

Cited by 13 publications

(18 citation statements)

References 9 publications

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“…It may be remarked that an additive analogue of Theorem 2.2 has been considered by Bapat, Lal and Pati (see [2]). For this, suppose we bidirect the edges of a tree and have weights w(a) for each arc a ∈ A.…”

Section: Elamentioning

confidence: 93%

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The product distance matrix of a tree and a bivariate zeta function of a graphs

Bapat¹,

Sivasubramanian²

2012

ELA

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Abstract. In this paper, the product distance matrix of a tree is defined and formulas for its determinant and inverse are obtained. The results generalize known formulas for the exponential distance matrix. When the number of variables are restricted to two, the bivariate analogue of the laplacian matrix of an arbitrary graph is defined. Also defined in this paper is a bivariate analogue of the Ihara-Selberg zeta function and its connection with the bivariate laplacian is shown. Finally, for connected graphs, there is a result connecting a partial derivative of the determinant of the bivariate laplacian and its number of spanning trees.

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Section: Elamentioning

confidence: 93%

“…For this, suppose we bidirect the edges of a tree and have weights w(a) for each arc a ∈ A. Define the distance between vertices i, j of the tree by replacing the product in (2.1) by the sum: [2]. The formula is fairly complicated, but when q e = q, t e = t for all e ∈ E(T ), it gets considerably simplified.…”

Section: Elamentioning

confidence: 99%

The product distance matrix of a tree and a bivariate zeta function of a graphs

Bapat¹,

Sivasubramanian²

2012

ELA

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“…The above expression gives a formula for the inverse of the distance matrix of a tree in terms of the Laplacian matrix. The determinant and the inverse of the distance matrix were also studied for bi-directed trees and weighted trees (for details, see [3,10]). In [2], similar results were studied for q-analogue of the distance ma- trix, which is a generalization of the distance matrix for a tree.…”

Section: Introductionmentioning

confidence: 99%

Distance Matrix of a Class of Completely Positive Graphs: Determinant and Inverse

2020

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AbstractA real symmetric matrix A is said to be completely positive if it can be written as BBt for some (not necessarily square) nonnegative matrix B. A simple graph G is called a completely positive graph if every matrix realization of G that is both nonnegative and positive semidefinite is a completely positive matrix. Our aim in this manuscript is to compute the determinant and inverse (when it exists) of the distance matrix of a class of completely positive graphs. We compute a matrix 𝒭 such that the inverse of the distance matrix of a class of completely positive graphs is expressed a linear combination of the Laplacian matrix, a rank one matrix of all ones and 𝒭. This expression is similar to the existing result for trees. We also bring out interesting spectral properties of some of the principal submatrices of 𝒭.

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“…The above expression gives formula for the inverse of distance matrix of a tree in terms of the Laplacian matrix. Several extensions and generalization of this result have been studied in [3,4,7,14,15,16,18,19]. The primary objective of these extensions is to define a matrix L called Laplacian-like matrix satisfying L1 = 0 and 1 t L = 0 and find the inverse of the distance matrix D(G) in the following form:…”

Section: Introductionmentioning

confidence: 99%

Distance Matrix of Weighted Cactoid-type Digraphs

Das¹,

Mohanty²

2020

Preprint

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A strongly connected digraph is called a cactoid-type, if each of its blocks is a digraph consisting of finitely many oriented cycles, sharing a common directed path. In this article, we find the formula for the determinant of the distance matrix for weighted cactoid-type digraphs and find its inverse, whenever it exists. We also compute the determinant of the distance matrix for a class of unweighted and undirected graphs consisting of finitely many cycles, sharing a common path.

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The distance matrix of a bidirected tree

Abstract: Abstract. A bidirected tree is a tree in which each edge is replaced by two arcs in either direction. Formulas are obtained for the determinant and the inverse of a bidirected tree, generalizing well-known formulas in the literature.

Cited by 13 publications

References 9 publications

The product distance matrix of a tree and a bivariate zeta function of a graphs

The product distance matrix of a tree and a bivariate zeta function of a graphs

Distance Matrix of a Class of Completely Positive Graphs: Determinant and Inverse

Distance Matrix of Weighted Cactoid-type Digraphs

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