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Cogrowth of Regular Graphs

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Regular graph
 covering tree
 amenable group
 random walk
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1992
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SUNY Plattsburgh Mathematics Faculty Work
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http://hdl.handle.net/20.500.12648/1109
Abstract
Let G be a d-regular graph and T the covering tree of G. We define a cogrowth constant of G in T and express it in terms of the first eigenvalue of the Laplacian on G. As a corollary, we show that the cogrowth constant is as large as possible if and only if the first eigenvalue of the Laplacian on G is zero. Grigorchuk's criterion for amenability of finitely generated groups follows.
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Northshield, S. (1992). Cogrowth of Regular Graphs. Proceedings of the American Mathematical Society, 116(1). http://doi.org/10.1090/S0002-9939-1992-1120509-0
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This article has been published in the September 1992 issue of Proceedings of the American Mathematical Society.
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