Energy and Laplacian energy of graphs (original) (raw)

Laplacian Energy of graphs in term of some graph Invariants

Let G be a simple graph with n vertices and m edges with  and  as maximum degree and vertex cover number of G Also let 1 ; 2 ; : : : ; n = 0 be the eigenvalues of the Laplacian matrix of graph G. The Laplacian energy of the graph G is dened as LE = LE(G) =... In this paper, we present some lower and upper bounds for LE of graph G in terms of n, the number of edges m and the maximum degree  and vertex cover number of  . Moreover, we obtain a relation between Laplacian energy and Laplacian-energy-like invariant of graphs.

On energy and Laplacian energy of graphs

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

On Laplacian energy in terms of graph invariants

For G being a graph with n vertices and m edges, and with Laplacian eigenvalues μ 1 ≥ μ 2 ≥ · · · ≥ μ n−1 ≥ μ n = 0, the Laplacian energy is defined as LE = n i=1 |μ i − 2m/n|. Let σ be the largest positive integer such that μ σ ≥ 2m/n. We characterize the graphs satisfying σ = n − 1.

Extremal Laplacian-energy-like invariant of graphs with given matching number

Electronic Journal of Linear Algebra, 2013

Let G be a graph of order n with Laplacian spectrum µ 1 ≥ µ 2 ≥ • • • ≥ µn. The Laplacian-energy-like invariant of graph G, LEL for short, is defined as: LEL(G) = n−1 k=1 √ µ k. In this note, the extremal (maximal and minimal) LEL among all the connected graphs with given matching number is determined. The corresponding extremal graphs are completely characterized with respect to LEL. Moreover a relationship between LEL and the independence number is presented in this note.

Some remarks on Laplacian eigenvalues and Laplacian energy of graphs

2010

Sažetak Suppose mu1\ mu_1 mu1, mu2\ mu_2 mu2,..., mun\ mu_n mun are Laplacian eigenvalues of a graph $ G .TheLaplacianenergyof. The Laplacian energy of .TheLaplacianenergyof G $ is defined as $ LE (G)=\ sum_ {i= 1}^ n|\ mu_i-2m/n| .Inthispaper,somenewboundsfortheLaplacianeigenvaluesandLaplacianenergyofsomespecialtypesofthesubgraphsof. In this paper, some new bounds for the Laplacian eigenvalues and Laplacian energy of some special types of the subgraphs of .Inthispaper,somenewboundsfortheLaplacianeigenvaluesandLaplacianenergyofsomespecialtypesofthesubgraphsof K_n $ are presented.

The energy of graph G is a graph invariant introduced by

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.

On energy and Laplacian energy of bipartite graphs

Let G be a bipartite graph of order n with m edges. The energy E(G) of G is the sum of the absolute values of the eigenvalues of the adjacency matrix A. In 1974, one of the present authors established lower and upper bounds for E(G) in terms of n, m, and det A. Now, more than 40 years later, we correct some details of this result and determine the extremal graphs. In addition, an upper bound on the Laplacian energy of bipartite graphs in terms of n, m, and the first Zagreb index is obtained, and the extremal graphs characterized.