Ehrhart theory, modular flow reciprocity, and the Tutte polynomial
Ehrhart theory, modular flow reciprocity, and the Tutte polynomial
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DOI：
10.1007/s00209-010-0782-6
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发表时间：
2009-07
影响因子：
0.8
通讯作者：
Felix Breuer;Raman Sanyal
中科院分区：
文献类型：
--
作者：
Felix Breuer;Raman Sanyal
给定一个有向图G，模流多项式计算G中无零流的个数。我们给出了一个描述的模流多项式的（开）Ehrhart多项式的格多面体。使用Ehrhart-Macdonald互易性，我们给出了一个组合解释的价值ofat负的论点，回答了贝克和Zaslavsky的问题（高等数学205：134-162，2006）。我们的结构扩展到$${\mathbb{Z}_{\l}}$$-张力，我们恢复斯坦利的色多项式的互易定理。结合流和张力的组合互易性陈述，我们给出了G的Tutte多项式G（x，y）的正赋值的计数解释。
Given an oriented graphG, the modular flow polynomialcounts the number of nowhere-zero-flows ofG. We give a description of the modular flow polynomial in terms of (open) Ehrhart polynomials of lattice polytopes. Using Ehrhart–Macdonald reciprocity we give a combinatorial interpretation for the values ofat negative arguments which answers a question of Beck and Zaslavsky (Adv Math 205:134–162, 2006). Our construction extends to $${\mathbb{Z}_{\l}}$$-tensions and we recover Stanley’s reciprocity theorem for the chromatic polynomial. Combining the combinatorial reciprocity statements for flows and tensions, we give an enumerative interpretation for positive evaluations of the Tutte polynomialtG(x,y) ofG.