Vertex-Colored Graphs, Bicycle Spaces and Mahler Measure
Document Type
Article
Publication Date
5-2016
Department
Mathematics
Abstract
The space C of conservative vertex colorings (over a field F) of a countable, locally finite graph G is introduced. When G is connected, the subspace C-0 of based colorings is shown to be isomorphic to the bicycle space of the graph. For graphs G with a cofinite free Z(d)-action by automorphisms, C is dual to a finitely generated module over the polynomial ring F[x(1)(+/- 1),..., x(d)(+/- 1)]. Polynomial invariants for this module, the Laplacian polynomials Delta k, k >= 0, are defined, and their properties are discussed. The logarithmic Mahler measure of Delta(0) is characterized in terms of the growth of spanning trees.
Publication Title
Journal of Knot Theory and Its Ramifications
Volume
25
Issue
6
Recommended Citation
Lamey, K. R.,
Silver, D. S.,
Williams, S. G.
(2016). Vertex-Colored Graphs, Bicycle Spaces and Mahler Measure. Journal of Knot Theory and Its Ramifications, 25(6).
Available at: https://aquila.usm.edu/fac_pubs/17472