Energy of Certain Planar Graphs
iir publications
Sign up for access to the world's latest research
checkGet notified about relevant papers
checkSave papers to use in your research
checkJoin the discussion with peers
checkTrack your impact
Abstract
The energy E(G) of a graph G is the sum of the absolute values of the eigenvalues of G of its adjacency matrix. The Laplacian energy LE(G) of a graph G is the sum of absolute values of its Laplacian eigenvalues. In this paper, we provide a MATLAB program, to calculate the energy and Laplacian energy of certain planar graphs namely n-regular caterpillar and Necklace.
Related papers
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
Laplacian energy of a graph
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.
Energy and 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
A bstrac t. Let G be an (n, m)-graph and µ1, µ2,..., µn its Laplacian eigenvalues. The Laplacian energy LE of G is defined as
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
Energy of Graph
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.
On the Laplacian energy of a graph
Czechoslovak Mathematical Journal, 2006
Some Notes on the Laplacian Energy of Extended Adjacency Matrix
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.
On Laplacian energy of graphs
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.
Energy of Some Simple Graphs: MATLAB Approach
In the present paper we have investigated MATLAB program to nd the energy of the graph. The energy of the Graph E(G) of G is the sum of absolute value of its eigen values. There are many research on energy of the graph, we have investigated the very new MATLAB program for nding energy of cycle, wheel and cyclic cubic graphs for n values and we consider example of all the graph for n 20. Keywords: MATLAB program, Energy Graph, cy- cle,wheel , cyclic cubic graph
On Path Laplacian Eigenvalues and Path Laplacian Energy of Graphs
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.
Laplacian Energy of a Graph with Self-Loops
Match Communications in Mathematical and in Computer Chemistry
The purpose of this paper is to extend the concept of Laplacian energy from simple graph to a graph with self-loops. Let G be a simple graph of order n, size m and GS is the graph obtained from G by adding σ self-loops. We define Laplacian energy of GS as LE(GS) = n i=1 µi(GS) − 2m+σ n where µ1(GS), µ2(GS),. .. , µn(GS) are eigenvalues of the Laplacian matrix of GS. In this paper some basic proprties of Laplacian eigenvalues and bounds for Laplacian energy of GS are investigated. This paper is limited to bounds in analogy with bounds of E(G) and LE(G) but with some significant differences, more sharper bounds can be found.
On the Randić energy of caterpillar graphs
MATCH Communications in Mathematical and in Computer Chemistry, 2021
A caterpillar graph T (p 1 ,. .. , p r) of order n = r + r i=1 p i , r ≥ 2, is a tree such that removing all its pendent vertices gives rise to a path of order r. In this paper we establish a necessary and sufficient condition for a real number to be an eigenvalue of the Randić matrix of T (p 1 ,. .. , p r). This result is applied to determine the extremal caterpillars for the Randić energy of T (p 1 ,. .. , p r) for cases r = 2 (the double star) and r = 3. We characterize the extremal caterpillars for r = 2. Moreover, we study the family of caterpillars T p, n−p−q−3, q of order n, where q is a function of p, and we characterize the extremal caterpillars for three cases: q = p, q = n − p − b − 3 and q = b, for b ∈ {1,. .. , n − 6} fixed. Some illustrative examples are included.
Some remarks on Laplacian eigenvalues and Laplacian energy of graphs
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.
Energy of Some New Graph
Advances in Mathematics: Scientific Journal, 2021
The energy of graph G is denoted by E(G), which is the sum of absolute values of its eigen values. Application of the energy graph is in chemistry to approximate the total π− electron energy of molecules. Moreover, we present results on the energy of a triangular book graph B(3, n), quadrilateral book graph B 4 n and restricted square of B (n,n) graph. 2020 Mathematics Subject Classification. 05C78.
On the edge energy of some specific graphs
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.
Energy of Certain Classes of Graphs Determined by Their Laplacian Degree Product Adjacency Spectrum
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.
Energy of line graphs
Linear Algebra and Its Applications, 2010
The energy of a graph is equal to the sum of the absolute values of its eigenvalues. The energy of a matrix is equal to the sum of its singular values. We establish relations between the energy of the line graph of a graph G and the energies associated with the Laplacian and signless Laplacian matrices of G.
A Note on Normalized Laplacian Energy of Graphs
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.
Laplacian Sum-Eccentricity Energy of a Graph
2017
We introduce the Laplacian sum-eccentricity matrix LS_e} of a graph G, and its Laplacian sum-eccentricity energy LS_eE=sum_{i=1}^n |eta_i|, where eta_i=zeta_i-frac{2m}{n} and where zeta_1,zeta_2,ldots,zeta_n are the eigenvalues of LS_e}. Upper bounds for LS_eE are obtained. A graph is said to be twinenergetic if sum_{i=1}^n |eta_i|=sum_{i=1}^n |zeta_i|. Conditions for the existence of such graphs are established.
On the Laplacian eigenvalues of a graph and Laplacian energy
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.
Loading Preview
Sorry, preview is currently unavailable. You can download the paper by clicking the button above.