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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Proc. Amer. Math. Soc. 137 (2009), 553-562 Request permission

Abstract:

The chord set of a function $f: \mathbb {R} \rightarrow \mathbb {R}$, denoted by $H(f)$, is the set of $r\in \mathbb {R}$ such that there exists $x\in \mathbb {R}$ with $f(x+r)=f(x)$. It is known that if $f$ is a continuous periodic function, then it has every chord, i.e. $H(f)=\mathbb {R}$. Equivalently, if $f$ is a real-valued Riemann-integrable function on the unit circle $C$ with $\int _C f =0$, then for any $r\in [0,1]$, there exists an arc $L$ of length $r$ such that $\int _L f=0$. In this paper, we formulate a definition of the chord set that gives way to generalizations on graphs. Given a connected finite graph $G$, we say $r\in H(G)$ if for any function $f \in L^1(G)$ with $\int _G f=0$ there exists a connected subset $A$ of size $r$ such that $\int _A f=0$. Among our results, we show that if $G$ has no vertex of degree 1, then $[0,l(G)] \subseteq H(G)$, where $l(G)$ is the length of the shortest closed path in $G$. Moreover, we show that if every vertex of a connected locally finite graph has even degree, then the graph has every chord.
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