2003/06/18 by Patrick McDonald, Robert Meyers, Robert A. Meyers · 2 citations
Mathematics · Computer Science · #Geometric and Algebraic Topology #Computational Geometry and Mesh Generation #Point processes and geometric inequalities
paper · pdf · doi:10.1090/s0002-9939-03-07123-5
Given a pair of planar isospectral, nonisometric polygons constructed as a quotient of the plane by a finite group, we construct an associated pair of planar isospectral, nonisometric weighted graphs. Using the natural heat operators on the weighted graphs, we associate to each graph a heat content. We prove that the coefficients in the small time asymptotic expansion of the heat content distinguish our isospectral pairs. As a corollary, we prove that the sequence of exit time moments for the natural Markov chains associated to each graph, averaged over starting points in the interior of the graph, provides a collection of invariants that distinguish isospectral pairs in general.