A Generalized Cheeger Inequality

TL;DR

This paper introduces a generalized Cheeger inequality relating the minimum generalized eigenvalue of Laplacian pairs to conductance measures between two graphs, extending classical spectral graph theory results.

Contribution

It establishes a new inequality connecting generalized conductance and Laplacian eigenvalues for pairs of graphs, broadening spectral graph analysis tools.

Findings

Derived a lower bound for the minimum generalized eigenvalue in terms of conductance measures.

Showed how to obtain a generalized cut from the eigenvector that meets the bound.

Connected the inequality to the Unique Games Conjecture, indicating its theoretical significance.

Abstract

The generalized conductance $\phi(G,H)$ between two graphs $G$ and $H$ on the same vertex set $V$ is defined as the ratio $$ \phi(G,H) = \min_{S\subseteq V} \frac{cap_G(S,\bar{S})}{ cap_H(S,\bar{S})}, $$ where $cap_G(S,\bar{S})$ is the total weight of the edges crossing from $S$ to $\bar{S}=V-S$. We show that the minimum generalized eigenvalue $\lambda(L_G,L_H)$ of the pair of Laplacians $L_G$ and $L_H$ satisfies $$ \lambda(L_G,L_H) \geq \phi(G,H) \phi(G)/8, $$ where $\phi(G)$ is the usual conductance of $G$. A generalized cut that meets this bound can be obtained from the generalized eigenvector corresponding to $\lambda(L_G,L_H)$. The inequality complements a recent proof that $\phi(G)$ cannot be replaced by $\Theta(\phi(G,H))$ in the above inequality, unless the Unique Games Conjecture is false.