On generalized distance energy of graphs

Mathematics, 2019

The generalized distance matrix D α ( G ) of a connected graph G is defined as D α ( G ) = α T r ( G ) + ( 1 − α ) D ( G ) , where 0 ≤ α ≤ 1 , D ( G ) is the distance matrix and T r ( G ) is the diagonal matrix of the node transmissions. In this paper, we extend the concept of energy to the generalized distance matrix and define the generalized distance energy E D α ( G ) . Some new upper and lower bounds for the generalized distance energy E D α ( G ) of G are established based on parameters including the Wiener index W ( G ) and the transmission degrees. Extremal graphs attaining these bounds are identified. It is found that the complete graph has the minimum generalized distance energy among all connected graphs, while the minimum is attained by the star graph among trees of order n.

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.

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.

Kragujevac J. …, 2009

1Department of Mathematics, Gogte Institute of Technology, Udyambag, Belgaum–590008, India (e-mails: [email protected], [email protected]) ... 2Faculty of Science, University of Kragujevac, PO Box 60, 34000 Kragujevac, Serbia (e-mail: [email protected])

Journal of Inequalities and Applications, 2018

Let G be a graph with n vertices and m edges. The term energy of a graph G was introduced by I. Gutman in chemistry due to its relevance to the total π-electron energy of a carbon compound. An analogous energy E D (G), called the distance energy, was defined by Indulal et al.

Mathematics, 2020

Given a simple connected graph G, let D ( G ) be the distance matrix, D L ( G ) be the distance Laplacian matrix, D Q ( G ) be the distance signless Laplacian matrix, and T r ( G ) be the vertex transmission diagonal matrix of G. We introduce the generalized distance matrix D α ( G ) = α T r ( G ) + ( 1 − α ) D ( G ) , where α ∈ [ 0 , 1 ] . Noting that D 0 ( G ) = D ( G ) , 2 D 1 2 ( G ) = D Q ( G ) , D 1 ( G ) = T r ( G ) and D α ( G ) − D β ( G ) = ( α − β ) D L ( G ) , we reveal that a generalized distance matrix ideally bridges the spectral theories of the three constituent matrices. In this paper, we obtain some sharp upper and lower bounds for the generalized distance energy of a graph G involving different graph invariants. As an application of our results, we will be able to improve some of the recently given bounds in the literature for distance energy and distance signless Laplacian energy of graphs. The extremal graphs of the corresponding bounds are also characterized.

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.

The distance energy of a graph G is defined as ED(G) = P |µi|, where µi is the i th eigenvalue of the distance matrix of G. In this paper, we express the distance spectra and distance energy of complete split graphs and graphs composed of two cliques joined by a matching. We also give some spectral properties of complete multipartite graphs. Finally, we identify structural and numerical conjectures on ED for graphs with number of vertices n and number of edges m are fixed.

2013

By a graph G = (V,E), we mean a finite undirected graph with neither loops nor multiple edges. The order and size of G are denoted by n = |V | and m = |E| respectively. For graph theoretic terminology we refer to Chartrand and Lesniak [7]. In Chapter 1, we collect some basic definitions and theorems on graphs which are needed for the subsequent chapters. The distance d(u, v) between two vertices u and v of a connected graph G is the length of a shortest u-v path in G. There are several distance related concepts and parameters such as eccentricity, radius, diameter, convexity and metric dimension which have been investigated by several authors in terms of theory and applications. An excellent treatment of various distances and distance related parameters are given in Buckley and Harary [6]. Let G = (V,E) be a graph. Let v ∈ V . The open neighborhood N(v) of a vertex v is the set of vertices adjacent to v. Thus N(v) = {w ∈ V : wv ∈ E}. The closed neighborhood of a vertex v, is the set...

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