A Cheeger Inequality for Size-Specific Conductance

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