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.

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