Departamento de Análise

Let T n be the set of all trees of diameter 3 and n vertices. We show that the Laplacian energy of any tree in T n is strictly between the Laplacian energy of the path P n and the star S n , partially proving the conjecture that this holds for any tree. We also give a total order by the Laplacian energy in T n . Moreover, we show that this order depends only on the algebraic connectivity of the tree: the Laplacian energy increases as the algebraic connectivity decreases in T n .

Let T n be the set of all trees of diameter 3 and n vertices. We show that the Laplacian energy of any tree in T n is strictly between the Laplacian energy of the path P n and the star S n , partially proving the conjecture that this holds for any tree. We also give a total order by the Laplacian energy in T n . Moreover, we show that this order depends only on the algebraic connectivity of the tree: the Laplacian energy increases as the algebraic connectivity decreases in T n .

Round-robin tournaments rank participants according to their results in all games, considering that all victories have the same value. However, our understanding is that victories against strong opponents have more value than those against weak opponents. Hence, such tournaments should consider this interpretation in their final rankings, using centrality measurements that evaluate the influence of an element based on its neighbours centralities. In this context, we use eigenvector and layer centralities to evaluate team performance in the 2014 Guanabara Cup, considering this understanding.

The aim of this article is to answer a question posed by Merris in European Journal of Combinatorics, 24(2003)413−430, about the possibility of finding split nonthreshold 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.

In this article, we characterize all signless Laplacian integral graphs (here called Q -integral graphs) in the following classes: complete split graphs, multiple complete split-like graphs, extended complete split-like graphs, multiple extended split-like graphs. All these graphs were defined by Hansen et al. [P. Hansen, H. Melot, D. Stevanović, Integral complete split graphs, Univ. Beograd, Publ. Elektrotehn. Fak. Ser. Mat. 13 (2002) 89-95]. Also, we characterize all Q -integral graphs of the form K n 1 ∨ (K n 2 ∪ K n 3 ), where K n i , i = 1, 2, 3, is a complete graph on n i vertices. These characterizations allow us to exhibit many infinite families of Q -integral graphs.

RESUMO A suavização da fronteira DEA foi proposta em 2002 para resolver o problema das múltiplas soluções ótimas para as DMUs extremo-eficientes na fronteira DEA clássica. Essa técnica substitui a fronteira original, linear por partes, por uma fronteira continuamente diferenciável, que passa pelas mesmas DMUs extremo-eficientes. Para escolher a fronteira que mais se aproxima da fronteira original, as funções objetivo dos diversos modelos de suavização existentes na literatura utilizam a topologia baseada no comprimento de arco (ou sua generalização multidimensional). Em 2014, foi proposto um modelo que utiliza outra função objetivo para a suavização da fronteira DEA, baseada na área (ou sua generalização multidimensional) das fronteiras. O presente trabalho estuda a topologia implicitamente utilizada nesse modelo, e também compara as características básicas das duas topologias da suavização. ABSTRACT Smooth DEA was first proposed in 2002 to solve the classic DEA problem of multiple optimal solutions for extreme-efficient DMUs. This technique replaces the original piecewise linear frontier by a new frontier with derivatives at all points that contains the same extreme-efficient DMUs. Most smooth models use the arc length based topology (or its multidimensional generalization) to determine the smooth frontier that is closest to the original DEA frontier. However, in 2014, the authors proposed a smooth model with a different objective function, based on the frontiersárea (or its multidimensional generalization). The present work studies the topology implicitly used in the new smooth model, and compares the basic characteristics from both topologies.

We characterize when a tree of diameter 4 has integer index and we provide examples of infinite families of non-integral trees with integer index. We also determine a tight upper bound for the index of any tree of diameter 4 based on its maximum degree. Moreover, we present a new infinite family of integral trees of diameter 4.