The cone of weighted graphs generated by triangles
Date
2016-05-05
Authors
Liu, Haggai
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Abstract
We investigate the problem of classifying complete edge weighted graphs on n vertices where each triangle has a nonnegative weight. We assign each edge an integer weight in a way that the edge weights are relatively prime. Such a graph G corresponds to a vector x with n 2 entries indexed by the edge set, E(G). The vecto, x, supports the cone of an inclusion matrix, W, of dimensions n 2 × n 3, with rows indexed by E(G) and columns indexed by the triangle of G. We wish to know whether or not x is a facet normal of this cone. In particular, we are interested in determining which weighted graphs corresponding to a facet normal. In general, for not very large n, there are many of these graphs corresponding to facet normals. Hence, we wish to count only the equivalence classes of facet normals under graph isomorphism. This problem is difficult in general, so it is best approached using computer software. To help with classifying facet normals, we present and implement an algorithm allowing us to find a facet normal at random.
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Keywords
combinatorial designs, geometry, graph theory, computing, complete edge weighted graphs, triangle, facet normals, isomorphism