On Laplacian energy of graphs

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

2018

We introduce the concept of Path Laplacian Matrix for a graph and explore the eigenvalues of this matrix. The eigenvalues of this matrix are called the path Laplacian eigenvalues of the graph. We investigate path Laplacian eigenvalues of some classes of graph. Several results concerning path Laplacian eigenvalues of graphs have been obtained.

Czechoslovak Mathematical Journal, 2006

MATCH Communications in Mathematical and in Computer Chemistry

Let G be a connected graph of order n with Laplacian eigenvalues μ 1 ≥μ 2 ≥⋯≥μ n-1 >μ n =0. The Laplacian energy of the graph G is defined as LE=LE(G)=∑ i=1 n |μ i -2m/n|. Upper bounds for LE are obtained, in terms of n and the number of edges m.

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.

… in Mathematical and in …, 2009

I. Gutman et al. have recently conjectured that the energy of a graph does not exceed its Laplacian energy. We disprove this conjecture by giving a few small counterexamples and, in addition, an infinite set of counterexamples. Nevertheless, we do show that the standard ...

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.

2009

The Laplacian energy of a graph G is defined as LE(G )= n=1 |λi − 2m n |, where λ1(G) ≥ λ2(G), ..., ≥ λn(G) = 0 are the Laplacian eigenvalues of the graph G. Some lower bounds for Laplacian energy of graphs are presented in this note.

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.

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.