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Transactions of the American Mathematical Society

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

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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On the period-two-property of the majority operator in infinite graphs
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by Gadi Moran PDF
Trans. Amer. Math. Soc. 347 (1995), 1649-1667 Request permission

Abstract:

A self-mapping $M:X \to X$ of a nonempty set $X$ has the Period-Two-Property (p2p) if ${M^2}x = x$ holds for every $M$-periodic point $x \in X$. Let $X$ be the set of all $\{ 0,1\}$-labelings $x:V \to \{ 0,1\}$ of the set of vertices $V$ of a locally finite connected graph $G$. For $x \in X$ let $Mx \in X$ label $v \in V$ by the majority bit that $x$ applies to its neighbors, retaining $\upsilon$’s $x$-label in case of a tie. We show that $M$ has the p2p if there is a finite bound on the degrees in $G$ and $\frac {1} {n}\log {b_n} \to 0$, where ${b_n}$ is the number of $\upsilon \in V$ at a distance at most $n$ from a fixed vertex ${\upsilon _0} \in V$.
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Additional Information
  • © Copyright 1995 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 347 (1995), 1649-1667
  • MSC: Primary 68R10; Secondary 68Q80, 68Q90, 90C35
  • DOI: https://doi.org/10.1090/S0002-9947-1995-1297535-1
  • MathSciNet review: 1297535