The determinant of the distance matrix of graphs with blocks at most bicyclic

Dratman, Ezequiel; Grippo, Luciano N.; Safe, Martín D.; Silva, Celso M. da; Del-Vecchio, Renata R.

doi:10.1016/j.laa.2020.11.008

Cited by 5 publications

(2 citation statements)

References 7 publications

(11 reference statements)

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“…We first recall C(n, m 1 , m 2 , • • • , m r ) is a undirected graph consisting of r cycles sharing a common path of length n. In this section, we compute the determinant of the distance matrix for a class of C(n, m 1 , m 2 , • • • , m r ). In literature, similar problems has been studied for the cases r = 1, 2 (for details see [3,8,9,10,15]) and we have shown that some of their results can be extended for r ≥ 3. We recall some results for the case r = 1, i.e.…”

Section: Distance Matrix Of Weighted Cactoid-type Digraphs: Determina...mentioning

confidence: 73%

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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Section: Distance Matrix Of Weighted Cactoid-type Digraphs: Determina...mentioning

confidence: 73%

Distance Matrix of Weighted Cactoid-type Digraphs

Das¹,

Mohanty²

2020

Preprint

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show abstract

“…In an early article, Graham and Pollack [10] presented a remarkable result, proving that the determinant of the distance matrix of a tree T on n vertices only depends on n, being equal to (−1) n−1 (n − 1)2 n−2 . More recently, formulas for the determinat of connected graphs on n vertices with n edges [1] (unicyclic graphs) and n + 1 edges [6] (bicyclic graphs) have been computed.…”

Section: Introductionmentioning

confidence: 99%

On the rank of the distance matrix of graphs

Dratman¹,

Grippo²,

Verónica³

et al. 2022

Preprint

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Let G be a connected graph with V (G) = {v1, . . . , vn}. The (i, j)entry of the distance matrix D(G) of G is the distance between vi and vj. In this article, using the well-known Ramsey's theorem, we prove that for each integer k ≥ 2, there is a finite amount of graphs whose distance matrices have rank k. We exhibit the list of graphs with distance matrices of rank 2 and 3. Besides, we study the rank of the distance matrices of graphs belonging to a family of graphs with their diameters at most two, the trivially perfect graphs. We show that for each η ≥ 1 there exists a trivially perfect graph with nullity η. We also show that for threshold graphs, which are a subfamily of the family of trivially perfect graphs, the nullity is bounded by one.

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A q-analogue of distance matrix of block graphs

Xing

2023

Advances in Applied Mathematics

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The determinant of the distance matrix of graphs with blocks at most bicyclic

Cited by 5 publications

References 7 publications

Distance Matrix of Weighted Cactoid-type Digraphs

Distance Matrix of Weighted Cactoid-type Digraphs

On the rank of the distance matrix of graphs

A q-analogue of distance matrix of block graphs

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