Eigenvalues of magnetic Laplacians on graphs and manifolds: Cheeger inequalities and Ramanujan property

Magnetic Laplacians arose from physical background. It is defined on a Riemannian manifold with a magnetic field. In this talk, we will discuss a corresponding concept of Cheeger type constants, which is a mixture of the isoperimetric ratio of the underlying manifold and the frustration index of the magnetic potentials, and the related (higher order) Cheeger inequalities. Those results on manifolds were inspired by our corresponding results for discrete magnetic Laplacians on finite graphs, for which Harary's structural balance theory for studying social networks plays an important role.

Discrecte Magnetic Laplacians on graphs also have important applications for constructing families of Ramanujan graphs. We will make this connection clear by explaining how the eigenvalues of a discrete magnetic Laplacian on the underlying graph G is related to the eigenvalues of the lifts of G.

This talk is based on the joint work with Carsten Lange, Norbert Peyerimhoff and Olaf Post, and the joint work with Norbert Peyerimhoff and Alina Vdovina.