Isoperimetric and Sobolev inequalities for magnetic graphs

TL;DR

This paper develops isoperimetric and Sobolev inequalities for magnetic graphs, introducing a new isoperimetric dimension concept and applying it to spectral bounds and graph products.

Contribution

It introduces the isoperimetric dimension for magnetic graphs and establishes inequalities linking isoperimetric and Sobolev properties in this context.

Findings

Signed Cheeger constant is additive under Cartesian products.

Lower bounds for eigenvalues of the magnetic Laplacian are derived.

Isoperimetric inequalities imply 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.