On the Non-Common Neighbourhood Energy of Graphs

Bulletin Classe des sciences mathematiques et natturalles

We introduce the concept of common-neighborhood energy E CN of a graph G and obtain an upper bound for E CN when G is strongly regular. We also show that E CN of several classes of graphs is less than the common-neighborhood energy of the complete graph K n .

Iranian Journal of Mathematical …, 2012

Let G be a simple graph with vertex set {v 1 , v 2 , . . . , vn}. The common neighborhood graph (congraph) of G, denoted by con(G), is the graph with vertex set {v 1 , v 2 , . . . , vn}, in which two vertices are adjacent if and only they have at least one common neighbor in the graph G. The basic properties of con(G) and of its energy are established.

Preprint, 2013

In this paper we introduce the concept of Energy of a Graph based on the minimum neighbourhood set of the graph. This energy is computed for some class of graphs.

International Journal of Mathematical Archive, 2016

L et G be connected graph with n vertices. The concept of degree sum matrix DS(G) of a simple graph G is introduced by H. S. Ramane et.al. [2]. And the degree sum energy E DS (G) [2] is defined by the sum of the absolute values of eigenvalues of the degree sum matrix DS(G) of G. The degree sum energy of a common neighborhood graph G [4] is defined by the sum of the absolute values of eigenvalues of the degree sum matrix of a common neighborhood graph DS[con(G)]. The terminal distance energy E T (G) of a graph [3] is defined by the sum of the absolute values of eigenvalues of the terminal distance matrix T(G) of a connected graph G. In this paper we modify upper bounds for the above defined energies.

Acta Universitatis Sapientiae, Informatica, 2020

Given a graph G = (V, E), with respect to a vertex partition 𝒫 we associate a matrix called 𝒫-matrix and define the 𝒫-energy, E𝒫 (G) as the sum of 𝒫-eigenvalues of 𝒫-matrix of G. Apart from studying some properties of 𝒫-matrix, its eigenvalues and obtaining bounds of 𝒫-energy, we explore the robust(shear) 𝒫-energy which is the maximum(minimum) value of 𝒫-energy for some families of graphs. Further, we derive explicit formulas for E𝒫 (G) of few classes of graphs with different vertex partitions.

Let G = (V, E) be a simple graph of order n with m edges. The energy of a graph G, denoted by E(G), is defined as the sum of the absolute values of all eigenvalues of G. The Laplacian energy of the graph G is defined as

Let G be a graph with n vertices and m edges. Let λ 1 ≥ λ 2 ≥ · · · ≥ λ n−1 ≥ λ n denote the eigenvalues of adjacency matrix A(G) of graph G . respectively. Then the Laplacian energy and the signless Laplacian energy of G are defined as

2019

This paper includes new bounds concepting the Seidel incidence energy. In the sequel, improved bounds about the Seidel Laplacian energy concerned with the edges and the vertices are established.

The main goal of this paper is to obtain some bounds for the normalized Laplacian energy of a connected graph. The normalized Laplacian energy of the line and para-line graphs of a graph are investigated. The relationship of the smallest and largest positive normalized Laplacian eigenvalues of graphs are also studied.

Annals of Pure and Applied Mathematics, 2018

Energy of a graph is an interesting parameter related to total π electron energy of the corresponding molecule. Recently Vaidya and Popat defined a pair of new graphs and obtained their energy in terms of the energy of original graph. In this paper we generalize the construction and obtain their energy. Also we discuss the spectrum of the first level thorn graph of a graph.