A Cheeger Inequality for Size-Specific Conductance

The $μ$-conductance measure proposed by Lovász and Simonovits is a size-specific conductance score that identifies the set with smallest conductance while disregarding those sets with volume smaller than a $μ$ fraction of the whole graph. Using $μ$-conductance enables us to study the network structures in new ways. In this manuscript we study a modified spectral cut for $μ$-conductance that is a natural relaxation of the integer program of $μ$-conductance and show that the optimum of this program has a two-sided Cheeger inequality with $μ$-conductance.