Vertex-Colored Graphs, Bicycle Spaces and Mahler Measure
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.
Journal of Knot Theory and Its Ramifications
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