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Title:
Multivariate flow and tension polynomials of graphs
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Abstract:
Flows and tensions of graphs are somewhat analogous to colorings, but one labels edges. Let's be more precise.Given a graph, orient it (i.e., give each edge an orientation) in some arbitrary but fixed way. A flow is a labelling of the edgessuch that at each node v, the sum of the labels at edges pointing towards v equals the sum of the labels at edges pointing away from v.A tension is a labelling of the edges such that the sum of the labels on any cycle of the graph (taken with signs given by theorientations of the edges of the cycle) is zero.One is typically interested in nowhere-zero flows and tensions, i.e., we're not allowed to use the label 0 anywhere.Some fairly recent theorems of Kochol and Chen say that if the flow/tension labelsare integers between -k and k, the number of nowhere-zero flows/tensions is a polynomial in k.Moreover, there are interpretations of these polynomials when they are evaluated at negative integers; these are reciprocity theorems due to Beck-Zaslavsky, and Chen-Stanley.We will try to generalize these counting functions to the multivariable case, i.e., for each edge e, we are allowed to use labelsbetween -ke and ke, for some given values ke.Now the flow and tension counting functions depend on several variables ke (one for each edge of the graph). We will study these counting functions.
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