Gain Graphs: Topology, Algebra, and Geometry

Suny At Binghamton, Binghamton NY

Investigators

Abstract

A "gain graph" is a graph together with a "gain function" that assigns to each edge an element of a group. If the group has order 2, the gain graph is a "signed graph". A gain graph has associated matroids, whose structure is more accessible than that of general matroids due to the relative simplicity of graphs. For this reason gain graphs have been helpful in a variety of geometry problems, for instance several related to root systems (since the classical root systems are described by signed graphs), as well as in optimization on generalized networks (also called "networks with gains"). The investigator plans to study what can be achieved in deeper understanding and application of gain graphs by "thickening" the edges. One way to do this is to replace each edge by its product with the boundary of a simplex of sufficiently high dimension to realize the action of the gain group, and then apply topology to the resulting thickened graph, for instance, calculating (co)homology and group actions on it. It is hoped that this will reveal striking new phenomena, perhaps analogously to the way the complexification of a real arrangement of hyperplanes has revealed phenomena sometimes entirely new and sometimes refining what was previously known. Think of a network, consisting of nodes connected by lines, in which each line has a "twist". In previous investigations of networks with twist (technical name: "gain graphs"), the lines were infinitely thin so the twists were purely theoretical. The investigator hopes to learn more about networks with twist by thickening the lines so the twists can have a real physical existence like the stripes on a barber pole and some candy canes or (more complexly) the strands of a braid. Then a whole new range of mathematical methods will become applicable so one can hope for new discoveries. The original reason for looking at twists was that they simplify some problems in geometry; they also turned out to be related to an economic optimization method. A detailed understanding of twisted networks is needed to find good solutions to problems like these. The hope for the thickening method is both to apply existing knowledge about twisted networks to simplify more mathematical problems and to find new techniques that reveal more about the twisted networks themselves. Like most mathematics, this is not aimed at immediate applications but at exciting new knowledge whose usefulness may become apparent years later.