On the adjacency matrix of a block graph

Bapat, R.B.; Roy, Souvik

doi:10.1080/03081087.2013.779268

Cited by 23 publications

(40 citation statements)

References 9 publications

(8 reference statements)

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“…A singular graph has a zero eigenvalue. Classifying singular graphs is a complicated problem in combinatorics [3,5,25]. In this article, we illuminate this problem with a number of examples with the new combinatorial implication.…”

Section: Example 1 the Digraphs Corresponding To The Matrices M And mentioning

confidence: 99%

“…In this article, we illuminate this problem with a number of examples with the new combinatorial implication. This procedure presents a simpli ed proof for the determinant of simple block graphs earlier given in [5]. These graph-theoretic representations would be useful in future investigations in matrix theory.…”

Section: Example 1 the Digraphs Corresponding To The Matrices M And mentioning

confidence: 99%

“…A component of G is a maximally connected subdigraph of G. A cut-vertex of G is a vertex whose removal results in an increase in the number of components in G. Now we de ne the idea of a block of a digraph, which plays a fundamental role in this article. It is already de ned in literature for simple graphs [5].…”

Section: A Digraph G = V(g) E(g) Is a Collection Of A Vertex Set V(gmentioning

confidence: 99%

“…For a simple graph G, when each of its blocks is a complete graph then G is called block graph [5]. The determinant of a block graph can be calculated from the theorem below, quoted from [5].…”

Section: Determinant Of Block Graphmentioning

confidence: 99%

“…Figure 1. Inspired by the work on simple block graphs [5], we propose a new technique for computing the characteristic and the permanent polynomials of a matrix. First of all, we derive a recursive expression for these polynomials of a matrix with respect to a pendant block in the corresponding digraph.…”

Section: De Nition 1 Block: a Block Is A Maximally Connected Subdigrmentioning

confidence: 99%

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On characteristic and permanent polynomials of a matrix

Singh

Bapat

2017

Special Matrices

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There is a digraph corresponding to every square matrix over C. We generate a recurrence relation using the Laplace expansion to calculate the characteristic and the permanent polynomials of a square matrix. Solving this recurrence relation, we found that the characteristic and the permanent polynomials can be calculated in terms of the characteristic and the permanent polynomials of some speci c induced subdigraphs of blocks in the digraph, respectively. Interestingly, these induced subdigraphs are vertex-disjoint and they partition the digraph. Similar to the characteristic and the permanent polynomials; the determinant and the permanent can also be calculated. Therefore, this article provides a combinatorial meaning of these useful quantities of the matrix theory. We conclude this article with a number of open problems which may be attempted for further research in this direction.

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Section: Example 1 the Digraphs Corresponding To The Matrices M And mentioning

confidence: 99%

Section: Example 1 the Digraphs Corresponding To The Matrices M And mentioning

confidence: 99%

Section: A Digraph G = V(g) E(g) Is a Collection Of A Vertex Set V(gmentioning

confidence: 99%

Section: Determinant Of Block Graphmentioning

confidence: 99%

Section: De Nition 1 Block: a Block Is A Maximally Connected Subdigrmentioning

confidence: 99%

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On characteristic and permanent polynomials of a matrix

Singh

Bapat

2017

Special Matrices

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Rank of a matrix of block graphs

Chen

Liao

2021

Int J of Quantum Chemistry

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The block graphs are good models in many real-world applications, such as molecular graph model for Polymers and windmill networks. A block graph is a graph in which every block is a complete graph. In this paper, we first prove that, the entries of distance matrix D G ð Þ of block graph G satisfy the famous Four point condition (4PC) and we define an matrixWe find that, if the block graph is a tree or each of its blocks is a triangle, thenand it is independent of the structure of block graphs G. Finally, we obtain the ℳ G index, Wiener index W G ð Þ and Wiener polarity index W B G ð Þ of isomers of heptane, respectively.

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Linear Time Algorithm to Check the Singularity of Block Graphs

Singh

Shaked-Monderer

Berman

2019

Lecture Notes in Computer Science

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A block graph is a graph in which every block is a complete graph. Let G be a block graph and let A(G) be its (0,1)-adjacency matrix. Graph G is called nonsingular (singular) if A(G) is nonsingular (singular). Characterizing nonsingular block graphs is an interesting open problem proposed by Bapat and Roy in 2013. In this article, we give a linear time algorithm to check whether a given block graph is singular or not.

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On the adjacency matrix of a block graph

Cited by 23 publications

References 9 publications

On characteristic and permanent polynomials of a matrix

On characteristic and permanent polynomials of a matrix

Rank of a matrix of block graphs

Linear Time Algorithm to Check the Singularity of Block Graphs

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