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 ringand for it polynomial invariants, the Laplacian polynomials ∆ k , k ≥ 0, are defined. Properties of the Laplacian polynomials are discussed. The logarithmic Mahler measure of ∆ 0 is characterized in terms of the growth of spanning trees.

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