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 fiedler's lemma and some applications

In a previous paper [M. Robbiano, E.A. Martins, and I. Gutman, Extending a theorem by Fiedler and applications to graph energy, MATCH Commun. Math. Comput. Chem. 64 (2010), pp. 145-156], a lemma by Fiedler was used to obtain eigenspaces of graphs, and applied to graph energy. In this article Fiedler's lemma is generalized and this generalization is applied to graph spectra and graph energy. © 2011 Taylor & Francis.

Necessary and sufficient conditions for a Hamiltonian graph

A graph is singular if the zero eigenvalue is in the spectrum of its 0-1 adjacency matrix A. If an eigenvector belonging to the zero eigenspace of A has no zero entries, then the singular graph is said to be a core graph. A ( k,t)-regular set is a subset of the vertices inducing a k -regular subgraph such that every vertex not in the subset has t neighbours in it. We consider the case when k=t which relates to the eigenvalue zero under certain conditions. We show that if a regular graph has a ( k,k )-regular set, then it is a core graph. By considering the walk matrix we develop an algorithm to extract ( k,k )-regular sets and formulate a necessary and sufficient condition for a graph to be Hamiltonian.

Spectra of graphs obtained by a generalization of the join graph operation

Taking a Fiedler’s result on the spectrum of a matrix formed from two symmetric matrices as a motivation, a more general result is deduced and applied to the determination of adjacency and Laplacian spectra of graphs obtained by a generalized join graph operation on families of graphs (regular in the case of adjacency spectra and arbitrary in the case of Laplacian spectra). Some additional consequences are explored, namely regarding the largest eigenvalue and algebraic connectivity.

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.

Majorantes para a ordem de subgrafos induzidos k-regulares

Muitos dos problemas de otimização em grafos reduzem-se à determinação de um subconjunto de vértices de cardinalidade máxima que induza um subgrafo k-regular. Uma vez que a determinação da ordem de um subgrafo induzido k-regular de maior ordem é, em geral, um problema NP-difícil, são deduzidos novos majorantes, a determinar em tempo polinomial, que em muitos casos constituam boas aproximações das respetivas soluções ótimas. Introduzem-se majorantes espetrais usando uma abordagem baseada em técnicas de programação convexa e estabelecem-se condições necessárias e suficientes para que sejam atingidos. Adicionalmente, introduzem-se majorantes baseados no espetro das matrizes de adjacência, laplaciana e laplaciana sem sinal. É ainda apresentado um algoritmo não polinomial para a determinação de umsubconjunto de vértices de umgrafo que induz umsubgrafo k-regular de ordem máxima para uma classe particular de grafos. Finalmente, faz-se um estudo computacional comparativo com vários majorantes e apresentam-se algumas conclusões.

On the Faria's inequality for the Laplacian and signless Laplacian spectra: a unified approach

Let p(G)p(G) and q(G)q(G) be the number of pendant vertices and quasi-pendant vertices of a simple undirected graph G, respectively. Let m_L±(G)(1) be the multiplicity of 1 as eigenvalue of a matrix which can be either the Laplacian or the signless Laplacian of a graph G. A result due to I. Faria states that mL±(G)(1) is bounded below by p(G)−q(G). Let r(G) be the number of internal vertices of G. If r(G)=q(G), following a unified approach we prove that mL±(G)(1)=p(G)−q(G). If r(G)>q(G) then we determine the equality mL±(G)(1)=p(G)−q(G)+mN±(1), where mN±(1) denotes the multiplicity of 1 as eigenvalue of a matrix N±. This matrix is obtained from either the Laplacian or signless Laplacian matrix of the subgraph induced by the internal vertices which are non-quasi-pendant vertices. Furthermore, conditions for 1 to be an eigenvalue of a principal submatrix are deduced and applied to some families of graphs.