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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Cogrowth of regular graphs
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by S. Northshield PDF
Proc. Amer. Math. Soc. 116 (1992), 203-205 Request permission

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
  • © Copyright 1992 American Mathematical Society
  • 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