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

TL;DR

This paper introduces a novel framework connecting orientations, lattice polytopes, and group arrangements to graph polynomials, providing new reciprocity laws and interpretations for the Tutte polynomial at specific points.

Contribution

It establishes reciprocity laws for integral and modular tension polynomials and offers a new interpretation of the Tutte polynomial at (1,0) within a unified subgroup arrangement framework.

Findings

Proves reciprocity law for integral tension polynomials

Establishes reciprocity law for modular tension polynomials

Interprets Tutte polynomial value at (1,0) as cut-equivalence classes of acyclic orientations

Abstract

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 $G$.