Isoperimetric and Sobolev inequalities for magnetic graphs

Abstract

We introduce a concept of isoperimetric dimension for magnetic graphs, that is, graphs where every edge is assigned a complex number of modulus one. In analogy with the classical case, we show that isoperimetric inequalities imply Sobolev inequalities on such graphs. As a first application, we show that the signed Cheeger constant behaves additively with respect to Cartesian products of graphs. Using heat kernel techniques, we also give lower bounds for the eigenvalues of the discrete magnetic Laplacian.