Nair Maria Maia de Abreu

A survey for the quadratic assignment problem

The quadratic assignment problem (QAP), one of the most difficult problems in the NP-hard class, models many real-life problems in several areas such as facilities location, parallel and distributed computing, and combinatorial data analysis. Combinatorial optimization problems, such as the traveling salesman problem, maximal clique and graph partitioning can be formulated as a QAP. In this paper, we present some of the most important QAP formulations and classify them according to their mathematical sources. We also present a discussion on the theoretical resources used to define lower bounds for exact and heuristic algorithms. We then give a detailed discussion of the progress made in both exact and heuristic solution methods, including those formulated according to metaheuristic strategies. Finally, we analyze the contributions brought about by the study of different approaches.

Bounds on the index of the signless Laplacian of a graph

A shock absorber apparatus connected in series relationship respective to the polished rod and bridle of a pumpjack unit. The shock absorber includes a housing within which there is supported a plurality of resilient packer members stacked in sandwiched relationship. The packer members have an axial passageway formed therethrough, with the passageway of each packer member being aligned with one another, so that the polished rod is telescopingly received therethrough. The bottom of the housing is supported by the bridle, while the polished rod is supported by a plate member which bears against the uppermost packer member. This places all of the resilient packer members in compression, with the force of the compression being proportional to the weight of the sucker rod.

The characteristic polynomial of the Laplacian of graphs in (a,b)-linear classes

In this work we deal with the characteristic polynomial of the Laplacian of a graph. We present some general results about the coefficients of this polynomial. We present families of graphs, for which the number of edges m is given by a linear function of the number of vertices n. In some of these graphs we can find certain coefficients of the above-named polynomial as functions just of n.

Constructing pairs of equienergetic and non-cospectral graphs

Abstract The energy of a simple graph G is the sum of the absolute values of the eigenvalues of its adjacency matrix. Two graphs of the same order are said to be equienergetic if they have the same energy. Several ways to construct equienergetic non-cospectral graphs of very large size can be found in the literature. The aim of this work is to construct equienergetic non-cospectral graphs of small size. In this way, we first construct several special families of such graphs, using the product and the cartesian product of complete graphs. Afterwards, we show how one can obtain new pairs of equienergetic non-cospectral graphs from the starting ones. More specifically, we characterize the connected graphs G for which the product and the cartesian product of G and K 2 are equienergetic non-cospectral graphs and we extend Balakrishnan’s result: For a non-trivial graph G , G ⊗ C 4 and G ⊗ K 2 ⊗ K 2 are equienergetic non-cospectral graphs, given in [R. Balakrishnan, The energy of a graph, Linear Algebra Appl. 387 (2004) 287–295].

Classes of quadratic assignment problem instances: isomorphism and difficulty measure using a statistical approach

In this work, we introduce the variance expression for quadratic assignment problem (QAP) costs. We also define classes of QAP instances, described by a common linear relaxation form. The use of the variance in these classes leads to the study of isomorphism and allows for a definition of a new difficulty index for QAP instances. This index is then compared to a classical measure found in the literature.

Variable Neighborhood Search for Extremal Graphs. XI. Bounds on Algebraic Connectivity

The algebraic connectivity a(G) of a graph G = (V, E) is the second smallest eigenvalue of its Laplacian matrix. Using the AutoGraphiX (AGX) system, extremal graphs for algebraic connectivity of G in function of its order n = |V| and size m = |E| are studied. Several conjectures on the structure of those graphs, and implied bounds on the algebraic connectivity, are obtained. Some of them are proved, e.g., if G ≠ K n

EXTREMAL ALGEBRAIC CONNECTIVITIES OF CERTAIN CATERPILLAR CLASSES AND SYMMETRIC CATERPILLARS

a\left( G \right) \leqslant \left\lfloor { - 1 + \sqrt {1 + 2m} } \right\rfloor

Uma revisão comentada das abordagens do problema quadrático de alocação

which is sharp for all m ≥ 2.