Title: Graph complexity and Mahler measure,
Abstract: The (torsion) complexity of a finite edge-weighted graph is defined to be the
order of the torsion subgroup of the abelian group presented by its Laplacian
matrix. When G is d-periodic (i.e., G has a free action of the rank-d free
abelian group by graph automorphisms, with finite quotient) the Mahler measure
of its Laplacian determinant polynomial is the growth rate of the complexity of
finite quotients of G. Lehmer's question, an open question about the roots of
monic integral polynomials, is equivalent to a question about the complexity
growth of edge-weighted 1-periodic graphs.