On The Isoperimetric Spectrum of Graphs

In this paper we introduce the k’th isoperimetric constant of a directed graph as the minimum of the mean outgoing normalized flows from a given set of k disjoint subsets of the vertex set of the graph. In this direction we show that the second isoperimetric constant in the 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 k’th isoperimetric constant and the number obtained by taking the minimum over all k-partitions. In this regard, we define the concept of a supergeometric graph by proving a Federer-Fleming-type theorem and analyzing the parameters through the corresponding functional definition. We also study the relationships of the isoperimetric spectrum to the classical spectrum of Laplacian eigenvalues, by proving a generalized Cheeger-type inequality as well as generalized Courant-Hilbert inequalities in the context of graph no-homomorphism theorems.