Adjacency Matrix of a Semigraph

doi:10.1016/j.endm.2017.11.037

Electronic Notes in Discrete Mathematics

Volume 63, December 2017, Pages 399-406

https://doi.org/10.1016/j.endm.2017.11.037 Get rights and content

Abstract

Semigraph was defined by Sampathkumar as a generalization of a graph. In this paper the adjacency matrix which represents semigraph uniquely and a characterization of such a matrix is obtained. An algorithm to construct the semigraph from a given square matrix, if semigraphical is given. Some properties of adjacency matrix of semigraph are studied. A sufficient condition for eigen values to be real is also obtained.

References (4)

R.B. Bapat
Graphs and Matrices
(2012)
Norman Biggs
Algebraic Graph Theory
(1974)

There are more references available in the full text version of this article.

Cited by (13)

On the existence of semigraphs and complete semigraphs with given parameters
2021, Ain Shams Engineering Journal
Citation Excerpt :
The idea of two or more vertices on an edge gives rise to a variety of generalizations of concepts in graph theory, when considered in semigraph theory. Many authors have proposed matrices associated with semigraphs and studied the properties of semigraphs with the help of those matrices [8–11]. Some results on the energy of a semigraph, making use of the adjacency matrix associated with a semigraph, are studied in [12].
E. Sampathkumar has generalized a graph to a semigraph by allowing an edge to have more than two vertices. Like in the case of graphs, a complete semigraph is a semigraph in which every two vertices are adjacent to each other. In this article, we have generalized a problem noted by Gauss in 1796 about triangular numbers and shown that it is the deciding factor of when a semigraph is complete.
Let P be a set with p elements and ${E_{1}, E_{2}, \dots, E_{q}}$ be a collection of subsets of P with $⋃_{i = 1}^{q} E_{i} = P$ . We derive an expression for the maximum value of the difference $(\sum_{j = 1}^{k} | E_{i_{j}} | - |⋃_{i = 1}^{k} E_{i_{j}}|)$ for $2 ⩽ k ⩽ q$ , where every two of the sets in the collection can have at most one element in common. We show that this result helps in answering the question of whether there exists a semigraph on the vertex set P having edges ${e_{1}, e_{2}, \dots, e_{q}}$ , where the set $E_{i}$ is the set of vertices on the edge $e_{i}, 1 ⩽ i ⩽ q$ . Combining the above two results, we characterize a complete semigraph.
Energy of a semigraph
2019, AKCE International Journal of Graphs and Combinatorics
Citation Excerpt :
Sampathkumar [6] generalized the definition of a graph to a semigraph in the following way. The adjacency matrix of a semigraph is defined as follows [7]. Nikiforov defined the energy of a general matrix (not necessarily square) as the summation of the singular values of that matrix.
Semigraph is a generalization of graph. We introduce the concept of energy in a semigraph in two ways, one, the matrix energy $E_{m}$ , as summation of singular values of the adjacency matrix of a semigraph, and the other, polynomial energy $E_{r e}$ , as energy of the characteristic polynomial of the adjacency matrix. We obtain some bounds for $E_{m}$ and show that $E_{m}$ is never a square root of an odd integer and $E_{r e}$ cannot be an odd integer. We investigate matrix energy of a partial semigraph and change in the matrix energy due to edge deletion.
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2023, arXiv
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2023, Global and Stochastic Analysis
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Adjacency Matrix of a Semigraph

Abstract

Graphs and Matrices

Algebraic Graph Theory