Applied Mathematics Department Colloquium - Gregory Berkolaiko

Gregory Berkolaiko, Department of Mathematics, Texas A&M University

Oscillation of graph eigenfunctions and its applications

Oscillation theory, originally due to Sturm, seeks to connect the number of sign changes of an eigenfunction of a self-adjoint operator to the label k of the corresponding eigenvalue.  Its applications run in both directions: knowing k, one may wish to estimate the zero set, or the topology of its complement, useful in clustering and partitioning problems.  Conversely, knowing an eigenvector (and thus the number of its sign changes), one may want to determine if it is the ground state, useful in the linear stability analysis of solutions to nonlinear equations. 

Within the setting of generalized graph Laplacians, Fiedler's theorem says that the k-th eigenvector of a tree (a graph without cycles) changes sign across exactly k-1 edges.  We present a formula for the number of sign changes on a general graph, which attributes the excess sign changes to the cycles in the graph and their intersections.

This result has many interesting connections.  First, it allows one to derive a simple formula for the effective mass tensor of a particular class of crystals (periodic lattices), namely the maximal abelian covers of finite graphs.  Second, it can be used to efficiently determine stability of a stationary solution on a coupled oscillator network, such as the non-uniform Kuramoto model for the synchronization of a network of electrical oscillators.  Finally, the determinant of the matrix which determines the excess sign changes is closely related to the graph's Kirchhoff polynomial (which counts the weighted spanning trees), hinting at connections to both Feynman amplitudes and matroids.

Based on a joint work with Jared Bronski (UI Urbana-Champaign) and Mark Goresky (IAS Princeton).