Optimal Grid Drawings of Complete Multipartite Graphs and an Integer Variant of the Algebraic Connectivity

How to draw the vertices of a complete multipartite graph $G$ on different points of a bounded $d$-dimensional integer grid, such that the sum of squared distances between vertices of $G$ is (i) minimized or (ii) maximized? For both problems we provide a characterization of the solutions. For the particular case $d=1$, our solution for (i) also settles the minimum-2-sum problem for complete bipartite graphs; the minimum-2-sum problem was defined by Juvan and Mohar in 1992. Weighted centroidal Voronoi tessellations are the solution for (ii). Such drawings are related with Laplacian eigenvalues of graphs. This motivates us to study which properties of the algebraic connectivity of graphs carry over to the restricted setting of drawings of graphs with integer coordinates.Comment: Appears in the Proceedings of the 26th International Symposium on Graph Drawing and Network Visualization (GD 2018

Spectral-based mesh segmentation

In design and manufacturing, mesh segmentation is required for FACE construction in boundary representation (BRep), which in turn is central for featurebased design, machining, parametric CAD and reverse engineering, among others -- Although mesh segmentation is dictated by geometry and topology, this article focuses on the topological aspect (graph spectrum), as we consider that this tool has not been fully exploited -- We preprocess the mesh to obtain a edgelength homogeneous triangle set and its Graph Laplacian is calculated -- We then produce a monotonically increasing permutation of the Fiedler vector (2nd eigenvector of Graph Laplacian) for encoding the connectivity among part feature submeshes -- Within the mutated vector, discontinuities larger than a threshold (interactively set by a human) determine the partition of the original mesh -- We present tests of our method on large complex meshes, which show results which mostly adjust to BRep FACE partition -- The achieved segmentations properly locate most manufacturing features, although it requires human interaction to avoid over segmentation -- Future work includes an iterative application of this algorithm to progressively sever features of the mesh left from previous submesh removal

Effect of Topological Structure and Coupling Strength in Weighted Multiplex Networks

International audienc

Addressing the envelope reduction of sparse matrices using a genetic programming system

Large sparse symmetric matrix problems arise in a number of scientific and engineering fields such as fluid mechanics, structural engineering, finite element analysis and network analysis. In all such problems, the performance of solvers depends critically on the sum of the row bandwidths of the matrix, a quantity known as envelope size. This can be reduced by appropriately reordering the rows and columns of the matrix, but for an N × N matrix, there are N! such permutations, and it is difficult to predict how each permutation affects the envelope size without actually performing the reordering of rows and columns. These two facts compounded with the large values of N used in practical applications, make the problem of minimising the envelope size of a matrix an exceptionally hard one. Several methods have been developed to reduce the envelope size. These methods are mainly heuristic in nature and based on graph-theoretic concepts. While metaheuristic approaches are popular alternatives to classical optimisation techniques in a variety of domains, in the case of the envelope reduction problem, there has been a very limited exploration of such methods. In this paper, a Genetic Programming system capable of reducing the envelope size of sparse matrices is presented and evaluated against four of the best-known and broadly used envelope reduction algorithms. The results obtained on a wide-ranging set of standard benchmarks from the Harwell–Boeing sparse matrix collection show that the proposed method compares very favourably with these algorithms