On nonsingular trees and a reciprocal eigenvalue property

Barik, Sasmita; Neumann, Michael; Pati, S.

doi:10.1080/03081080600792897

Cited by 55 publications

(21 citation statements)

References 6 publications

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“…The following description of the inverse of the adjacency matrix of a connected h-graph was given in [9,10]. We follow the convention that sum over an empty class is zero.…”

Section: Preliminariesmentioning

confidence: 99%

On some graphs which possess inverses

Panda

Pati

2015

Linear and Multilinear Algebra

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A graph G is nonsingular if its adjacency matrix A(G) is nonsingular. A nonsingular graph G is said to have an inverse G + if A(G) −1 is signature similar to a nonnegative matrix. An h-graph G is a bipartite graph with a unique perfect matching M. We introduce the 'even'ness of a nonmatching edge and show that under certain conditions a connected h-graph has an inverse if and only if (G − E)/M is bipartite, where E is the set of all even nonmatching edges. This extends some known results, providing us with a larger class of graphs possessing inverses.

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“…The following description of the inverse of the adjacency matrix of a connected h-graph was given in [9,10]. We follow the convention that sum over an empty class is zero.…”

Section: Preliminariesmentioning

confidence: 99%

On some graphs which possess inverses

Panda

Pati

2015

Linear and Multilinear Algebra

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“…A corona tree is a corona graph which is a tree. The following description of the inverse of the adjacency matrix of a connected, bipartite graph with a unique perfect matching is a restatement of results contained in [1], [2]. We follow the convention that a sum over an empty set is zero.…”

Section: Unicycyclic Graphs Which Possess Bicyclic Inversesmentioning

confidence: 99%

Unicyclic graphs with bicyclic inverses

Panda

2017

Czech.Math.J.

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“…In [1], it is proved that a tree T has property (SR) if and only if T = T 1 • K 1 , for some tree T 1 . This was strengthened in [2] where it is proved that a tree has property (R) if an only if it has property (SR).…”

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confidence: 99%

Unicyclic graphs with strong reciprocal eigenvalue property

Barik¹,

Pati²,

Sarma³

2008

ELA

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Abstract.A graph G is bipartite if and only if the negative of each eigenvalue of G is also an eigenvalue of G. It is said that a graph has property (R), if G is nonsingular and the reciprocal of each of its eigenvalues is also an eigenvalue. Further, if the multiplicity of an eigenvalue equals that of its reciprocal, the graph is said to have property (SR). The trees with property (SR) have been recently characterized by Barik, Pati and Sarma. Barik, Neumann and Pati have shown that for trees the two properties are, in fact, equivalent. In this paper, the structure of a unicyclic graph with property (SR) is studied. It has been shown that such a graph is bipartite and is a corona (unless it has girth four). In the case it is not a corona, it is shown that the graph can have one of the three specified structures. Families of unicyclic graphs with property (SR) having each of these specific structures are provided.

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On nonsingular trees and a reciprocal eigenvalue property

Cited by 55 publications

References 6 publications

On some graphs which possess inverses

On some graphs which possess inverses

Unicyclic graphs with bicyclic inverses

Unicyclic graphs with strong reciprocal eigenvalue property

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