A generalized Cheeger inequality

The generalized conductance phi(G, H) between two weighted graphs G and H on the same vertex set V is defined as the ratio phi(G, H) = min S subset of V capG(S, S) capH(S, S over line ), where capG(S, S) is the total weight of the edges crossing from vertex set S subset of V to S over line = V - S. We show that the minimum generalized eigenvalue lambda(LG, LH) of the pair of Laplacians LG and LH satisfies phi(G, H) > lambda(LG, LH) > phi(G, H)phi(G)/16, where phi(G) is the standard conductance of G. A generalized cut that meets this bound can be obtained from the generalized eigenvector corresponding to lambda(LG, LH).(c) 2023 Elsevier Inc. All rights reserved.