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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Isospectral polygons, planar graphs and heat content


Authors: Patrick McDonald and Robert Meyers
Journal: Proc. Amer. Math. Soc. 131 (2003), 3589-3599
MSC (2000): Primary 58J50, 58J65
Published electronically: June 18, 2003
MathSciNet review: 1991773
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Abstract: 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.


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Additional Information

Patrick McDonald
Affiliation: Department of Mathematics, New College of Florida, Sarasota, Florida 34243
Email: ptm@virtu.sar.usf.edu

Robert Meyers
Affiliation: The Courant Institute of Mathematical Sciences, New York, New York 10276-0907
Email: rjm243@nyu.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-03-07123-5
PII: S 0002-9939(03)07123-5
Keywords: Isospectral pairs, Dirichlet spectrum, heat content
Received by editor(s): May 10, 2002
Published electronically: June 18, 2003
Communicated by: Jozef Dodziuk
Article copyright: © Copyright 2003 American Mathematical Society