𝒫-Energy of 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.

2021

The energy of a graph G, denoted by E(G), is defined as the sum of the absolute values of all eigenvalues of G. In this paper, lower and upper bounds for energy in some of the graphs are established, in terms of graph invariants such as the number of vertices, the number of edges, and the number of closed walks.

ITM Web of Conferences

In this paper, we obtain some upper and lower bounds for the spectral radius of some special matrices such as maximum degree, minimum degree, Randic, sum-connectivity, degree sum, degree square sum, first Zagreb and second Zagreb matrices of a simple connected graph G by the help of matrix theory. We also get some upper bounds for the corresponding energies of G.

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.

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

Match-communications in Mathematical and in Computer Chemistry, 2020

Let G be a simple undirected graph with n vertices, m edges, adjacency matrix A, largest eigenvalue ρ and nullity κ. The energy of G, E(G) is the sum of its singular values. In this work lower bounds for E(G) in terms of the coefficient of μκ in the expansion of characteristic polynomial, p(μ) = det (μI −A) are obtained. In particular one of the bounds generalizes a lower bound obtained by K. Das, S. A. Mojallal and I. Gutman in 2013 to the case of graphs with given nullity. The bipartite case is also studied obtaining in this case, a sufficient condition to improve the spectral lower bound 2ρ. Considering an increasing sequence convergent to ρ a convergent increasing sequence of lower bounds for the energy of G is constructed.

2013

The energy of a graph G, denoted by E(G), is defined as the sum of the absolute values of all eigenvalues of G . In this paper we present some new upper bounds for E(G) in terms of number of vertices, number of edges, clique number, minimum degree, and the first Zagreb index.

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

Match-communications in Mathematical and in Computer Chemistry, 2020

Let G be a simple undirected graph with n vertices and m edges. The energy of G, E(G) corresponds to the sum of its singular values. This work obtains lower bounds for E(G) where one of them generalizes a lower bound obtained by Mc Clelland in 1971 to the case of graphs with given nullity. An extension to the bipartite case is given and, in this case, it is shown that the lower bound 2 √ m is improved. The equality cases are characterized. Moreover, a simple lower bound that considers the number of edges and the diameter of G is derived. A simple lower bound, which improves the lower bound 2 √ n− 1, for the energy of trees with n vertices and diameter d is also obtained. 1 Notation and Preliminaries In this work we deal with an (n,m)-graph G which is an undirected simple graph with vertex set V (G) and edge set E (G) of cardinality n and m, respectively. As usual we denote the adjacency matrix of G by A = A(G). The eigenvalues of G are the eigenvalues of A (see e.g. [5, 6]). Its eig...

arXiv: Spectral Theory, 2019

Let $R$ be a Hermitian matrix. The energy of $R$, $\mathcal{E}(R)$, corresponds to the sum of the absolute values of its eigenvalues. In this work it is obtained two lower bounds for $\mathcal{E}(R).$ The first one generalizes a lower bound obtained by Mc Clellands for the energy of graphs in $1971$ to the case of Hermitian matrices and graphs with a given nullity. The second one generalizes a lower bound obtained by K. Das, S. A. Mojallal and I. Gutman in 2013 to symmetric non-negative matrices and graphs with a given nullity. The equality cases are discussed. These lower bounds are obtained for graphs with $m$ edges and some examples are provided showing that, some obtained bounds are incomparable with the known lower bound for the energy $2\sqrt{m}$. Another family of lower bounds are obtained from an increasing sequence of lower bounds for the spectral radius of a graph. The bounds are stated for singular and non-singular graphs.