119 research outputs found
Cycle and Circle Tests of Balance in Gain Graphs: Forbidden Minors and Their Groups
We examine two criteria for balance of a gain graph, one based on binary
cycles and one on circles. The graphs for which each criterion is valid depend
on the set of allowed gain groups. The binary cycle test is invalid, except for
forests, if any possible gain group has an element of odd order. Assuming all
groups are allowed, or all abelian groups, or merely the cyclic group of order
3, we characterize, both constructively and by forbidden minors, the graphs for
which the circle test is valid. It turns out that these three classes of groups
have the same set of forbidden minors. The exact reason for the importance of
the ternary cyclic group is not clear.Comment: 19 pages, 3 figures. Format: Latex2e. Changes: minor. To appear in
Journal of Graph Theor
A New Algorithm in Geometry of Numbers
A lattice Delaunay polytope P is called perfect if its Delaunay sphere is the
only ellipsoid circumscribed about P. We present a new algorithm for finding
perfect Delaunay polytopes. Our method overcomes the major shortcomings of the
previously used method. We have implemented and used our algorithm for finding
perfect Delaunay polytopes in dimensions 6, 7, 8. Our findings lead to a new
conjecture that sheds light on the structure of lattice Delaunay tilings.Comment: 7 pages, 3 figures; Proceedings of ISVD-07, International Symposium
on Voronoi diagrams in Science and Engineering held in July of 2007 in Wales,
U
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