On The Isoperimetric Spectrum of Graphs and Its Approximations

TL;DR

This paper explores higher isoperimetric numbers of directed graphs, establishing their properties, differences from traditional measures, and proposing spectral inequalities and algorithms for approximation.

Contribution

It introduces a new definition of the nth mean isoperimetric constant for directed graphs, proves its fundamental properties, and connects it to spectral inequalities and algorithmic approaches.

Findings

Second mean isoperimetric constant coincides with classical Cheeger constant.

Identifies fundamental differences between nth isoperimetric constants and n-partition minima.

Provides spectral inequalities and algorithms related to graph isoperimetry.

Abstract

In this paper we consider higher isoperimetric numbers of a (finite directed) graph. In this regard we focus on the $n$th mean isoperimetric constant of a directed graph as the minimum of the mean outgoing normalized flows from a given set of $n$ disjoint subsets of the vertex set of the graph. We show that the second mean isoperimetric constant in this general setting, coincides with (the mean version of) the classical Cheeger constant of the graph, while for the rest of the spectrum we show that there is a fundamental difference between the $n$th isoperimetric constant and the number obtained by taking the minimum over all $n$-partitions. In this direction, we show that our definition is the correct one in the sense that it satisfies a Federer-Fleming-type theorem, and we also define and present examples for the concept of a supergeometric graph as a graph whose mean isoperimetric constants are attained on partitions at all levels. Moreover, considering the ${\bf NP}$-completeness of the isoperimetric problem on graphs, we address ourselves to the approximation problem where we prove general spectral inequalities that give rise to a general Cheeger-type inequality as well. On the other hand, we also consider some algorithmic aspects of the problem where we show connections to orthogonal representations of graphs and following J.~Malik and J.~Shi ($2000$) we study the close relationships to the well-known $k$-means algorithm and normalized cuts method.