Vertex-Colored Graphs, Bicycle Spaces and Mahler Measure

TL;DR

This paper introduces a new algebraic framework for analyzing vertex-colored graphs using bicycle spaces and Mahler measure, linking graph invariants to algebraic and dynamical properties.

Contribution

It defines the space of conservative vertex colorings, relates it to bicycle spaces, and introduces the Laplacian polynomial and Mahler measure as new invariants for graphs with group actions.

Findings

The space of conservative colorings is isomorphic to the bicycle space.

The Laplacian polynomial's Mahler measure relates to spanning tree growth.

Properties of the Laplacian polynomial are characterized for graphs with free Z^d-actions.

Abstract

The space C of conservative vertex colorings (over a field F) of a countable, locally finite graph G is introduced. The subspace of based colorings is shown to be isomorphic to the bicycle space of the graph. For graphs G with a free Z^d-action by automorphisms, C is a finitely generated module over the polynomial ring F[Z^d], and for this a polynomial invariant, the Laplacian polynomial, is defined. Properties of this polynomial are discussed. The logarithmic Mahler measure of the Laplacian polynomial is characterized in terms of the growth of spanning trees of G.