The first Cheeger constant of a simplex

TL;DR

This paper determines the exact value of the first Cheeger constant of a simplex for almost all sizes, revealing it equals n/3 except when n is a power of two, where it approaches n/3 asymptotically.

Contribution

It proves the first Cheeger constant of a simplex equals n/3 for all n not a power of two, and approaches n/3 for powers of two, expanding previous bounds.

Findings

Exact value h_1(Δ^{[n]})=n/3 for non-powers of two.

h_1(Δ^{[n]}) approaches n/3 for powers of two.

Method involves graph-theoretic analysis of staircase graphs.

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 $k\geq 0$. 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, that \[\lceil n/3\rceil\geq h_1(\Delta^{[n]})\geq n/3, \textrm{ for all } n,\] 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 \[h_1(\Delta^{[n]})=n/3, \textrm{ whenever }n\textrm{ is not a power of }2.\] So the sharp equality holds on a set whose density goes to $1$. Second, we show that \[h_1(\Delta^{[n]})=n/3+O(1/n),\textrm{ when }n\textrm{ is a power of }2.\] 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 {\it 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.