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Orientations, Lattice Polytopes, and Group Arrangements I: Chromatic and Tension Polynomials of Graphs
Beifang Chen
Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong
mabfchen@ust.hk
Annals of Combinatorics 13 (4) pp.425-452 December, 2009
AMS Subject Classification: 05A99, 05B35, 05C15, 06A07, 52B20, 52B40
Abstract:
This is the first one of a series of papers on association of orientations, lattice polytopes, and group arrangements to graphs. The purpose is to interpret the integral and modular tension olynomials 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 a viewpoint leads to the ollowing 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.
Keywords: subgroup arrangements, hyperplane arrangements, orientations, acyclic orientations, oriented cuts, cut-equivalence relation, coloring polytopes, tension polytopes, chromatic polynomial, tension polynomial, characteristic polynomial, Tutte polynomial, reciprocity law

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