Energy of line graphs

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. © 2010 Elsevier B.V. All rights reserved.

A lower bound for the energy of symmetric matrices and graphs

The energy of a symmetric matrix is the sum of the absolute values of its eigenvalues. We introduce a lower bound for the energy of a symmetric partitioned matrix into blocks. This bound is related to the spectrum of its quotient matrix. Furthermore, we study necessary conditions for the equality. Applications to the energy of the generalized composition of a family of arbitrary graphs are obtained. A lower bound for the energy of a graph with a bridge is given. Some computational experiments are presented in order to show that, in some cases, the obtained lower bound is incomparable with the well known lower bound $2\sqrt{m}$, where $m$ is the number of edges of the graph.

Bounds for the signless laplacian energy

The energy of a graph G is the sum of the absolute values of the eigenvalues of the adjacency matrix of G. The Laplacian (respectively, the signless Laplacian) energy of G is the sum of the absolute values of the differences between the eigenvalues of the Laplacian (respectively, signless Laplacian) matrix and the arithmetic mean of the vertex degrees of the graph. In this paper, among some results which relate these energies, we point out some bounds to them using the energy of the line graph of G. Most of these bounds are valid for both energies, Laplacian and signless Laplacian. However, we present two new upper bounds on the signless Laplacian which are not upper bounds for the Laplacian energy. © 2010 Elsevier Inc. All rights reserved.

A generalization of the Hoffman-Lovász upper bound on the independence number of a regular graph

A family of quadratic programming problems whose optimal values are upper bounds on the independence number of a graph is introduced. Among this family, the quadratic programming problem which gives the best upper bound is identified. Also the proof that the upper bound introduced by Hoffman and Lovász for regular graphs is a particular case of this family is given. In addition, some new results characterizing the class of graphs for which the independence number attains the optimal value of the above best upper bound are given. Finally a polynomial-time algorithm for approximating the size of the maximum independent set of an arbitrary graph is described and the computational experiments carried out on 36 DIMACS clique benchmark instances are reported.

Bethe graphs attached to the vertices of a connected graph: a spectral approach

A weighted Bethe graph $B$ is obtained from a weighted generalized Bethe tree by identifying each set of children with the vertices of a graph belonging to a family $F$ of graphs. The operation of identifying the root vertex of each of $r$ weighted Bethe graphs to the vertices of a connected graph $\mathcal{R}$ of order $r$ is introduced as the $\mathcal{R}$-concatenation of a family of $r$ weighted Bethe graphs. It is shown that the Laplacian eigenvalues (when $F$ has arbitrary graphs) as well as the signless Laplacian and adjacency eigenvalues (when the graphs in $F$ are all regular) of the $\mathcal{R}$-concatenation of a family of weighted Bethe graphs can be computed (in a unified way) using the stable and low computational cost methods available for the determination of the eigenvalues of symmetric tridiagonal matrices. Unlike the previous results already obtained on this topic, the more general context of families of distinct weighted Bethe graphs is herein considered.