Some Notes on the Laplacian Energy of Extended Adjacency Matrix

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.

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.

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.

Journal of Mathematics, 2022

In this study, we investigate the Laplacian degree product spectrum and corresponding energy of four families of graphs, namely, complete graphs, complete bipartite graphs, friendship graphs, and corona products of 3 and 4 cycles with a null graph.

2020

In this paper we define e-Adjacency matrix A_e(S) and e-Laplacian matrix of a Semigraph L_e(S). Also discuss some results of eigenvalues of these matrices. We define e-Energy of Semigraph E_e(S) using eigenvalues of its e-adjacency matrix and e- Laplacian energy of Semigraph LE_e (S) using eigenvalues of its e-Laplacian matrix. We investigate relation between e-energy E_e(S) and e-Laplacian energy LE_e(S) for regular Semigraphs.

The paper is essentially a survey of known results about the spectrum of the Laplacian matrix of graphs with special emphasis on the second smallest Laplacian eigenvalue λ 2 and its relation to numerous graph invariants, including connectivity, expanding properties, isoperimetric number, maximum cut, independence number, genus, diameter, mean distance, and bandwidth-type parameters of a graph. Some new results and generalizations are added.

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.

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.

Czechoslovak Mathematical Journal, 2010

The Laplacian spectral radius of a graph is the largest eigenvalue of the associated Laplacian matrix. In this paper, we determine those graphs which maximize the Laplacian spectral radius among all bipartite graphs with (edge-)connectivity at most k. We also characterize graphs of order n with k cut-edges, having Laplacian spectral radius equal to n.

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.

Linear Algebra and its Applications, 2013

The second largest Laplacian eigenvalue of a graph is the second largest eigenvalue of the associated Laplacian matrix. In this paper, we study extremal graphs for the extremal values of the second largest Laplacian eigenvalue and the Laplacian separator of a connected graph, respectively. All simple connected graphs with second largest Laplacian eigenvalue at most 3 are characterized. It is also shown that graphs with second largest Laplacian eigenvalue at most 3 are determined by their Laplacian spectrum. Moreover, the graphs with maximum and the second maximum Laplacian separators among all connected graphs are determined.

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.

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.

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.

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.