TITLE

# The First Cheeger Constant of a Simplex

AUTHOR(S)
Kozlov, Dmitry
PUB. DATE
November 2017
SOURCE
Graphs & Combinatorics;Nov2017, Vol. 33 Issue 6, p1543
SOURCE TYPE
DOC. TYPE
Article
ABSTRACT
The coboundary expansion generalizes the classical graph expansion to the case of the general simplicial complexes, and allows the definition of the higher-dimensional Cheeger constants $$h_k(X)$$ for an arbitrary simplicial complex X, and any $$k\ge 0$$ . In this paper we investigate the value of $$h_1(\Delta ^{[n]})$$ -the first Cheeger constant of a simplex with n vertices. It is known, due to the pioneering work of Meshulam and Wallach [12], that and that the equality $$h_1(\Delta ^{[n]})=n/3$$ is achieved when n is divisible by 3. Here we expand on these results. First, we show that So the sharp equality holds on a set whose density goes to 1. Second, we show that In other words, as n goes to infinity, the value $$h_1(\Delta ^{[n]})-n/3$$ is either 0 or goes to 0 very rapidly. Our methods include recasting the original question in purely graph-theoretic language, followed by a detailed investigation of a specific graph family, the so-called staircase graphs. These are defined by associating a graph to every partition, and appear to be especially suited to gain information about the first Cheeger constant of a simplex.
ACCESSION #
126308051

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