Hardy-Muckenhoupt Bounds for Laplacian Eigenvalues

May 1, 2019 (GHC 8102)

We present two graph quantities $\psi(G,S)$ and $\psi_2(G)$ which give constant factor estimates to the Dirichlet and Neumann eigenvalues of the Laplacian, $\lambda(G,S)$ and $\lambda_2(G)$, respectively. These quantities are “electrical" relaxations of the sparsest cut on a graph and our main theorem gives an “electrical” alternative to Cheeger’s inequality. Along the way we will talk about spring-mass systems and Muckenhoupt’s weighted Hardy inequality.