2016/02/08 by Silver, Daniel S., Williams, Susan G.
#05C10. 37B10 #57M25 #82B20 #Combinatorics (math.CO) #FOS: Mathematics #Geometric Topology (math.GT)
paper · doi:10.48550/arxiv.1602.02797
The complexity of a finite connected graph is its number of spanning trees; for a non-connected graph it is the product of complexities of its connected components. If G is an infinite graph with cofinite free \mathbb Zd-symmetry, then the logarithmic Mahler measure m(Δ) of its Laplacian polynomial Δ is the exponential growth rate of the complexity of finite quotients of G. It is bounded below by m(Δ(\mathbb Gd)), where \mathbb Gd is the grid graph of dimension d. The growth rates m(Δ(\mathbb Gd)) are asymptotic to log 2d as d tends to infinity. If m(Δ(G))≠ 0, then m(Δ(G)) ≥ log 2. An application to determinant growth rates of families of alternating links arising from planar graphs is given.