Laplacian Sum-Eccentricity Energy of a Graph

Czechoslovak Mathematical Journal, 2006

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.

Linear Algebra and its Applications, 2010

Let k be a natural number and let G be a graph with at least k vertices. A.E. Brouwer conjectured that the sum of the k largest Laplacian eigenvalues of G is at most e(G) + k+1 2 , where e(G) is the number of edges of G. We prove this conjecture for k = 2. We also show that if G is a tree, then the sum of the k largest Laplacian eigenvalues of G is at most e(G) + 2k − 1.

2010

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

Discrete Mathematics, 2014

Let G be a graph with n vertices and m edges. Also let µ 1 , µ 2 ,. .. , µ n−1 , µ n = 0 be the eigenvalues of the Laplacian matrix of graph G.

2020

Let G be a simple, connected graph on the vertex set V(G) and the edge set E(G). For the degree of the vertex denoted by , the maximum degree is denoted by and the minimum degree is denoted by . If and are adjacent, then it is represented by . The adjacency matrix is a symmetric square matrix that determines the corner pairs in a graph. Let denote the eigenvalues of adjacency matrix. The greatest eigenvalue is said to as the spectral radius of the graph G. The energy of graph G is defined as . The Laplacian matrix of a graph G is represented by where is the degree matrix. The degree matrix is the diagonal matrix formed by the degree of each point belonging to G. The Laplacian eigenvalues are real. The graph laplacian energy is described by = with edges and vertices.

2017

Let G = (V,E) be a simple graph. The energy of G is the sum of absolute values of the eigenvalues of its adjacency matrix A(G). In this paper we consider the edge energy of G (or energy of line of G) which is defined as the absolute values of eigenvalues of edge adjacency matrix of G. We study the edge energy of specific graphs.

2019

Let $G$ be a graph. The Laplacian matrix of $G$ is $L(G)=D(G)-A(G)$, where $D(G)=diag(d(v_{1}),\ldots , d(v_{n}))$ is a diagonal matrix and $d(v)$ denotes the degree of the vertex $v$ in $G$ and $A(G)$ is the adjacency matrix of $G$. Let $G_1$ and $G_2$ be two (unicyclic) graphs. We study the multiplicity of the Laplacian eigenvalue $2$ of $G=G_1\odot G_2$ where the graphs $G_1$ or $G_2$ may have perfect matching and Laplacian eigenvalue $2$ or not. We initiate the Laplacian characteristic polynomial of $G_1$, $G_2$ and $G=G_1\odot G_2$. It is also investigated that Laplacian eigenvalue $2$ of $G=G_1\odot G_2$ for some graphs $G_1$ and $G_2$ under the conditions.

Match-communications in Mathematical and in Computer Chemistry, 2020

Let G be a simple undirected graph with n vertices, m edges, adjacency matrix A, largest eigenvalue ρ and nullity κ. The energy of G, E(G) is the sum of its singular values. In this work lower bounds for E(G) in terms of the coefficient of μκ in the expansion of characteristic polynomial, p(μ) = det (μI −A) are obtained. In particular one of the bounds generalizes a lower bound obtained by K. Das, S. A. Mojallal and I. Gutman in 2013 to the case of graphs with given nullity. The bipartite case is also studied obtaining in this case, a sufficient condition to improve the spectral lower bound 2ρ. Considering an increasing sequence convergent to ρ a convergent increasing sequence of lower bounds for the energy of G is constructed.

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.

Linear Algebra and its Applications, 2008

For a graph G and a real α / = 0, we study the graph invariant s α (G)-the sum of the αth power of the non-zero Laplacian eigenvalues of G. The cases α = 2, 1 2 and −1 have appeared in different problems. Here we establish some properties for s α with α / = 0, 1. We also discuss the cases α = 2, 1 2 .

For a graph G with vertex set V(G) = {v 1

Linear Algebra and its Applications, 2015

Let G be a simple graph with n vertices, m edges, maximum degree Δ, average degree d = 2m n , clique number ω having Laplacian eigenvalues μ 1 , μ 2 ,. .. , μ n−1 , μ n = 0. For k (1 ≤ k ≤ n), let S k (G) = k i=1 μ i and let σ (1 ≤ σ ≤ n − 1) be the number of Laplacian eigenvalues greater than or equal to average degree d. In this paper, we obtain a lower bound for S ω−1 (G) and an upper bound for S σ (G) in terms of m, Δ, σ and clique number ω of the graph. As an application, we obtain the stronger bounds for the Laplacian energy LE(G) = n i=1 |μ i − d|, which improve some well known earlier bounds.

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 be a graph with n vertices and m edges. Let λ 1 , λ 2 , . . . , λ n be the eigenvalues of the adjacency matrix of G, and let µ 1 , µ 2 , . . . , µ n be the eigenvalues of the Laplacian matrix of G. An earlier much studied quantity E(G) = n i=1 |λ i | is the energy of the graph G. We now define and investigate the Laplacian energy as LE(G) = n i=1 |µ i − 2m/n|. There is a great deal of analogy between the properties of E(G) and LE(G), but also some significant differences.