The classical Cheeger inequality relates the edge conductance $\phi$ of the graph to the second smallest eigenvalue $\lambda_2$ of the Laplacian matrix. Recently, Olesker-Taylor and Zanetti discovered the Cheeger-type inequality $\psi^2 / \log |V|. \lesssim \lambda_2^* \lesssim \psi$ The vertex expansion $\psi$ of the graph $G=(V,E)$ and the maximally reweighted second smallest eigenvalue $\lambda_2^*$ of the Laplacian matrix are Connect.
In this work, we first refine the result to $\psi^2 / \log d \lesssim \lambda_2^* \lesssim \psi$. where $d$ is the maximum degree of $G$. Make an expansion forecast. The improved results also hold for weighted vertex expansion, answering an open question by Olesker-Taylor and Zanetti. Based on this connection, we develop a new spectral theory for vertex extension. It turns out that some interesting generalizations of the Cheeger inequality involving edge conductance and eigenvalues have similarities involving vertex expansion and reweighted eigenvalues. These include analogs of Trevisan’s results for bipartiteness, analogs of higher-order Cheeger’s inequalities, and analogs of improved Cheeger’s inequalities.
Finally, inspired by this relationship, we present negative evidence for the $0/1$ polytope edge evolution conjecture by Mihail and Vazirani. Constructs $0/1$-polytopes with very bad vertex expansion of the graph. This means that the fastest mixing time to a uniform distribution at the vertices of these $0/1$ polytopes is approximately linear with graph size. Although this does not provide a counterexample to the conjecture, it contrasts with known positive results that have proven polylogarithmic mixing times for uniform distributions at the vertices of subclasses of the $0/1$ polytope.
