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Cogrowth of regular graphs


Author: S. Northshield
Journal: Proc. Amer. Math. Soc. 116 (1992), 203-205
MSC: Primary 60J15; Secondary 05C05, 43A05
DOI: https://doi.org/10.1090/S0002-9939-1992-1120509-0
MathSciNet review: 1120509
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Abstract: Let $ \mathcal{G}$ be a $ d$-regular graph and $ T$ the covering tree of $ \mathcal{G}$. We define a cogrowth constant of $ \mathcal{G}$ in $ T$ and express it in terms of the first eigenvalue of the Laplacian on $ \mathcal{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 $ \mathcal{G}$ is zero. Grigorchuk's criterion for amenability of finitely generated groups follows.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1992-1120509-0
Keywords: Regular graph, covering tree, amenable group, random walk
Article copyright: © Copyright 1992 American Mathematical Society

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