Ağırlıklı Enerji İçin Bazı Sınırlar (original) (raw)
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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
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.
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
Bounds for the energy of weighted graphs
Discrete Applied Mathematics, 2019
Let G be a simple connected graph with n vertices and m edges. Let W (G) = (G, w) be the weighted graph corresponding to G. Let λ 1 , λ 2 ,. .. , λ n be the eigenvalues of the adjacency matrix A(W (G)) of the weighted graph W (G). The energy E(W (G)) of a weighted graph W (G) is defined as the sum of absolute value of the eigenvalues of W (G). In this paper, we obtain upper bounds for the energy E(W (G)), in terms of the sum of the squares of weights of the edges, the maximum weight, the maximum degree d 1 , the second maximum degree d 2 and the vertex covering number τ of the underlying graph G. As applications to these upper bounds we obtain some upper bounds for the energy (adjacency energy), the extended graph energy, the Randić energy and the signed energy of the connected graph G. We also obtain some new families of weighted graphs where the energy increases with increase in weights of the edges.
The energy of graphs and matrices
Journal of Mathematical Analysis and Applications, 2007
Given a complex m n matrix A; we index its singular values as 1 (A) 2 (A) ::: and call the value E (A) = 1 (A) + 2 (A) + ::: the energy of A; thereby extending the concept of graph energy, introduced by Gutman. Let 2 m n; A be an m n nonnegative matrix with maximum entry , and kAk 1 n. Extending previous results of Koolen and Moulton for graphs, we prove that
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.
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.
On the Laplacian energy of graphs
Let G be a simple graph with n vertices and m edges with ∆ and δ as maximum and minimum degrees of G. Also let µ1 , µ2 , . . . , µn = 0 be the eigenvalues of the Laplacian matrix of graph G. The Laplacian energy of the graph G is defined as
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.