Vertex-Colored Graphs, Bicycle Spaces and Mahler Measure
Combinatorics
2015-10-15 v3 Dynamical Systems
Geometric Topology
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.
Keywords
Cite
@article{arxiv.1408.6570,
title = {Vertex-Colored Graphs, Bicycle Spaces and Mahler Measure},
author = {Kalyn R. Lamey and Daniel S. Silver and Susan G. Williams},
journal= {arXiv preprint arXiv:1408.6570},
year = {2015}
}
Comments
Version 3 strengthens theorem 7.7, adds some references and makes other small changes