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Graph Complexity and Mahler Measure

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Dan Silver (South Alabama)

Abstract: In this joint work with Susan G. Williams we define the (torsion) complexity of a finite edge-weighted graph 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 $Z^d$-action 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. Lehmerâ€™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. Connections with knot theory are discussed.