Mathematische Zeitschrift

, Volume 270, Issue 1–2, pp 1–18 | Cite as

Ehrhart theory, modular flow reciprocity, and the Tutte polynomial

Open Access


Given an oriented graph G, the modular flow polynomial \({\phi_G(k)}\) counts the number of nowhere-zero \({\mathbb{Z}_k}\)-flows of G. 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 of \({\phi_G}\) at 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 polynomial tG(x, y) of G.


Nowhere-zero flows Modular flow polynomial Reciprocity theorems Lattice-point counting Ehrhart theory Tutte polynomial 

Mathematics Subject Classification (2000)

Primary 05C31 Graph polynomials Secondary 05C21 Flows in graphs 52B20 Lattice polytopes 52C35 Arrangements of points, flats, hyperplanes 


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Copyright information

© The Author(s) 2010

Authors and Affiliations

  1. 1.Institut für Mathematik, Freie Universität BerlinBerlinGermany
  2. 2.Department of MathematicsUniversity of California, BerkeleyBerkeleyUSA

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