HARARY EQUIENERGETIC GRAPHS

2004

The distance energy E D (G) of a graph G is defined as the sum of the absolute values of the eigenvalues of the distance matrix of G. The graphs G 1 and G 2 are said to be distance equienergetic (D-equienergetic) if E D (G 1 ) = E D (G 2 ) . In this paper we obtain the eigenvalues of the distance matrix of the join of two graphs whose diameter is less than or equal to 2, and construct pairs of non D-cospectral, D-equienergetic graphs on n vertices for all n ≥ 9 .

The energy of a graph is the sum of the absolute values of its eigenvalues. Let G and L 2 (G) denote the complement and the second iterated line graph, respectively, of the graph G. If G 1 and G 2 are two regular graphs, both on n vertices, both of degree r ≥ 3 , then L 2 (G 1) and L 2 (G 2) have equal energies, equal to (nr − 4)(2r − 3) − 2 .

arXiv: Combinatorics, 2019

Energy of a simple graph $G$, denoted by $\mathcal{E}(G)$, is the sum of the absolute values of the eigenvalues of $G$. Two graphs with the same order and energy are called equienergetic graphs. A graph $G$ with the property $G\cong \overline{G}$ is called self-complementary graph, where $\overline{G}$ denotes the complement of $G$. Two non-self-complementary equienergetic graphs $G_1$ and $G_2$ satisfying the property $G_1\cong \overline{G_2}$ are called complementary equienergetic graphs. Recently, Ramane et al. [Graphs equienergetic with their complements, MATCH Commun. Math. Comput. Chem. 82 (2019) 471-480] initiated the study of the complementary equienergetic regular graphs and they asked to study the complementary equienergetic non-regular graphs. In this paper, by developing some computer codes and by making use of some software like Nauty, Maple and GraphTea, all the complementary equienergetic graphs with at most 10 vertices as well as all the members of the graph class $\...

2007

The D-eigenvalues of a graph G are the eigenvalues of its distance matrix D , and the D-energy E D (G) is the sum of the absolute values of its D-eigenvalues. Two graphs are said to be D-equienergetic if they have the same D-energy. In this note we obtain bounds for the distance spectral radius and D-energy of graphs of diameter 2. Pairs of equiregular D-equienergetic graphs of diameter 2, on p = 3t + 1 vertices are also constructed.

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.

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.

2013

Two graphs are said to be equienergetic if their energies are equal. In the paper MATCH Commun. Math. Comput. Chem. 61 (2009) 451–461 the concept of almost-equienergetic graphs was put forward, based on the observation that in some cases the (non-zero) difference between the energies of two graphs is very small. We now estimate the minimal value of this difference.

2015

This dissertation brings together two important concepts in graph theory the energy of a graph and the complete graph. The energy of a graph is the sum of the absolute values its eigenvalues, and originated from the determination of the sum of π -electron energy in a molecule represented by a molecular graphi.e. a graph where the vertices represent atoms and the edges bonds between atoms. Important theorems, such as the Lovazs and Lollipop theorems, are used to find eigenvalues of classes of graphs while analytic methods are used to determine simplified expressions of the energy of classes of graphs. As a result of the investigation, in the literature, of the difference of the energy of two graphs G and H, on the same number n of vertices, we adapted this idea by making one of the graphs the complete graph. This premise is based on the fact that the complete graph is a very important and well-studied class of graphs. Since the complete graph does not have the largest energy of all g...

2010

The main purposes of this paper are to introduce and investigate the Harary energy and Harary Estrada index of a graph. In addition we establish upper and lower bounds for these new energy and index separately.

2017

Let G be a graph without an isolated vertex, the normalized Laplacian matrix L̃(G) is defined as L̃(G) = D− 1 2L(G)D− 1 2 , where D is a diagonal matrix whose entries are degree of vertices of G. The eigenvalues of L̃(G) are called as the normalized Laplacian eigenvalues of G. In this paper, we obtain the normalized Laplacian spectrum of two new types of join graphs. In continuing, we determine the integrality of normalized Laplacian eigenvalues of graphs. Finally, the normalized Laplacian energy and degree Kirchhoff index of these new graph products are derived. c ⃝ 2017 IAUCTB. All rights reserved.