#### Date of Original Version

12-19-2014

#### Type

Working Paper

#### Abstract or Description

The generalized conductance ϕ(G,H) between two graphs G and H on the same vertex set Vis defined as the ratio ϕ(G,H)=minS⊆VcapG(S,S¯)capH(S,S¯),

where capG(S,S¯) is the total weight of the edges crossing from S to S¯=V−S. We show that the minimum generalized eigenvalue λ(LG,LH) of the pair of Laplacians LG and LH satisfies λ(LG,LH)≥ϕ(G,H)ϕ(G)/8,

where ϕ(G) is the usual conductance of G. A generalized cut that meets this bound can be obtained from the generalized eigenvector corresponding to λ(LG,LH). The inequality complements a recent proof that ϕ(G) cannot be replaced by Θ(ϕ(G,H)) in the above inequality, unless the Unique Games Conjecture is false.