Vertex-Colored Graphs, Bicycle Spaces and Mahler Measure

Spanning Trees and Mahler Measure

“…We apply the above ideas to graphs G with cofinite free Z d -symmetry. As in [15], define the coloring module C to be the finitely presented module over the ring R d with presentation matrix equal to the n × n Laplacian matrix L of G. The Laplacian polynomial ∆ arises as the 0th elementary divisor of C.…”

“…We call the limit in Theorem 3.6 the complexity growth rate of G, and denote it by γ(G). Its relationship to the thermodynamic limit or bulk limit defined for a wide class of lattice graphs is discussed in [15]. We briefly repeat the idea in order to state Corollary 3.9.…”

“…When the graph G admits a cofinite free Z d -symmetry (see definition below), a precise connection with algebraic dynamics was made in [15]. For such graphs a finitely generated "coloring module" over the ring of Laurent polynomials Z[x ±1 1 , .…”

“…The Laplacian polynomial ∆ is the determinant of L. It is well defined up to multiplication by units of the ring R d . Examples can be found in [15].…”

“…(3) Theorem 3.6 was established in [15] with the weaker limit superior rather than an ordinary limit. The stronger result will follow from analytical remarks in [7] related to Mahler measure.…”