e-Adjacency matrix and e-Laplacian matrix of semigraph

In general, graphs are associated with different types of matrices to study their structural properties. As a result, the study of spectrum of graphs become an important area of research in graph theory. Various types of energies have been discussed according to the type of matrix under consideration. In semigraphs, the adjacency between vertices is defined in many ways. This opens a broad scope to study the different energies of semigraphs. In this paper, two types of adjacency matrices, namely e-adjacency matrix and n-adjacency matrix are considered to study the respective energies of the most fundamental semigraph known as linear semigraph.

2021

In this paper, some properties of semi-regular graphs have been studied. The energy of graphs has many mathematical properties, which are being investigated for some of the semi-regular graphs. Also, the Laplacian Energy of these types of the graph has been defined has also been studied. We give examples of semi-regular graphs, describe the barbell class, and describe how the property of semi regularity relates to other properties of graphs.

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.

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.

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

Czechoslovak Mathematical Journal, 2006

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.

A b s t r a c t. Let G be an (n, m)-graph and µ 1 , µ 2 , . . . , µ n its Laplacian eigenvalues. The Laplacian energy LE of G is defined as

Journal for Research in Applied Sciences and Biotechnology

By given the adjacency matrix, laplacian matrix of a graph we can find the set of eigenvalues of graph in order to discussed about the energy of graph and laplacian energy of graph. (i.e. the sum of eigenvalues of adjacency matrix and laplacian matrix of a graph is called the energy of graph) and the laplacian energy of graph is greater or equal to zero for any graph and is greater than zero for every connected graph with more or two vertices (i.e. the last eigenvalues of laplacian matrix is zero), according to several theorems about the energy of graph and the laplacian energy of graph that are described in this work; I discussed about energy of graph, laplacian energy of graph and comparing them here.