Nair Maria Maia de Abreu

We determine a closed-form expression for the fifth characteristic coefficient of the power of a path. To arrive at this result, we establish the number of 4-cycles in the graph by means of their structural properties. The method developed might be applied to other well-structured graph classes in order to count 4-cycles or modified to count cycles of different length.

In this paper, trees with fixed diameter and any number of vertices are investigated. A subclass of trees with diameter 2k is introduced, the diameter path trees (dp-trees). Two subclasses of dp-trees are defined in which we characterize the elements that maximize the algebraic connectivity. Also, it is proved that if any tree maximizes the algebraic connectivity over all trees with diameter 2k then it is a dp-tree. For such trees, a bound for the degrees of their vertices is given. In the case of the odd diameter, 2k-1, we show that P2k is the only tree that maximizes the algebraic connectivity.

This paper gives a tight upper bound on the spectral radius of the signless Laplacian of graphs of given order and clique number. More precisely, let G be a graph of order n, let A be its adjacency matrix, and let D be the diagonal matrix of the row-sums of A. If G has clique number r, then the largest eigenvalue q(G) of the matrix Q=A+D satisfies q(G)<= 2(1-1/r)n. If G is a complete regular r-partite graph, then equality holds in the above inequality. This result confirms a conjecture of Hansen and Lucas.

The second smallest Laplacian eigenvalue of a graph G is called algebraic connectivity, denoted a(G). The ordering of trees via this graph invariant is frequently studied in the literature. In this paper, we present a new invariant, the Internal Degree Sequence (IDS), that also supports an accurate evaluation of the connectivity of trees. We compare the IDS with a(G) for all elements in six classes of trees known to have the largest algebraic connectivity and we show that the IDS provides a strict total ordering of the elements of these classes. This result is also proved for a subclass of trees of diameter 4

The aim of this article is to answer a question posed by Merris in European Journal of Combinatorics, 24 (2003) pp. 413 − 430, about the possibility of finding split non-threshold graphs that are Laplacian integral, i.e. graphs for which the eigenvalues of the corresponding Laplacian matrix are integers. Using Kronecker products, balanced incomplete block designs, and solutions to certain Diophantine equations, we show how to build infinite families of these graphs.

O Problema Quadrático de Alocação, PQA, pode ser abordado através de uma relaxação na forma do Problema de Alocação Linear, PAL. Introduzimos um poset (conjunto parcialmente ordenado) no conjunto das soluções lineares que nos permite comparar também os custos das soluções do problema quadrático, sem o conhecimento prévio das matrizes que definem seus exemplares. Construímos um algoritmo polinomial capaz de determinar pares de permutações livremente comparáveis, conceito apresentado neste trabalho. Provamos um teorema que garante que os custos associados a tais permutações preservam a ordem dada pelo número de inversões das mesmas. Associando soluções do problema a permutações, testes empíricos são apresentados, visando a validação do número de inversões como um parâmetro de referência para a qualidade das soluções. Este trabalho é uma extensão do artigo [RA01] publicado nos anais do XXXIII SBPO, em CD-ROM, 1277-1287, Sobrapo, ILTC, 2001.

Este texto é um pequeno e merecido tributo ao Professor Nelson Maculan Filho por ocasião do seu 60º aniversário, em reconhecimento às suas grandes contribuições à Universidade brasileira. Como amigos, colaboradores e ex-alunos, ficamos sensibilizados e honrados com o convite feito pelo Professor Jayme Luiz Szwarcfiter para escrever este artigo. Carpe diem, querido amigo Maculan!