Orientations, lattice polytopes, and group arrangements I: Chromatic and tension polynomials of graphs

Details Orientations, lattice polytopes, and group arrangements I: Chromatic and tension polynomials of graphs Orientations, lattice polytopes, and group arrangements I: Chromatic and tension polynomials of graphs

2007-06-22

arxiv.org/abs/0706.3273v1

Mathematics

Combinatorics

25 pages, 3 figures

Scientific paper

This is the first one of a series of papers on association of orientations, lattice polytopes, and abelian group arrangements to graphs. The purpose is to interpret the integral and modular tension polynomials of graphs at zero and negative integers. The whole exposition is put under the framework of subgroup arrangements and the application of Ehrhart polynomials. Such viewpoint leads to the following main results of the paper: (i) the reciprocity law for integral tension polynomials; (ii) the reciprocity law for modular tension polynomials; and (iii) a new interpretation for the value of the Tutte polynomial $T(G;x,y)$ of a graph $G$ at $(1,0)$ as the number of cut-equivalence classes of acyclic orientations on $G$.

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