Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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

The 2024 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
Proc. Amer. Math. Soc. 116 (1992), 203-205
DOI: https://doi.org/10.1090/S0002-9939-1992-1120509-0

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.
References
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Bibliographic 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