Lattice Points in Orthotopes and a Huge Polynomial Tutte Invariant of Weighted Gain Graphs

TL;DR

This paper introduces a new Tutte polynomial invariant for weighted gain graphs with lattice-ordered gain groups, connecting graph colorings, lattice points, and matrix counts, extending classical invariants and providing explicit formulas.

Contribution

It develops a general dichromatic polynomial for weighted gain graphs that unifies several classical and recent graph invariants and explores its applications to lattice point counting and matrix enumeration.

Findings

A new Tutte invariant for weighted gain graphs is established.

The polynomial includes several classical graph invariants as special cases.

Explicit formulas are derived for counting lattice points and matrices in specified regions.

Abstract

A gain graph is a graph whose edges are orientably labelled from a group. A weighted gain graph is a gain graph with vertex weights from an abelian semigroup, where the gain group is lattice ordered and acts on the weight semigroup. For weighted gain graphs we establish basic properties and we present general dichromatic and forest-expansion polynomials that are Tutte invariants (they satisfy Tutte's deletion-contraction and multiplicative identities). Our dichromatic polynomial includes the classical graph one by Tutte, Zaslavsky's two for gain graphs, Noble and Welsh's for graphs with positive integer weights, and that of rooted integral gain graphs by Forge and Zaslavsky. It is not a universal Tutte invariant of weighted gain graphs; that remains to be found. An evaluation of one example of our polynomial counts proper list colorations of the gain graph from a color set with a gain-group action. When the gain group is Z^d, the lists are order ideals in the integer lattice Z^d, and there are specified upper bounds on the colors, then there is a formula for the number of bounded proper colorations that is a piecewise polynomial function of the upper bounds, of degree nd where n is the order of the graph. This example leads to graph-theoretical formulas for the number of integer lattice points in an orthotope but outside a finite number of affinographic hyperplanes, and for the number of n x d integral matrices that lie between two specified matrices but not in any of certain subspaces defined by simple row equations.