REU Project

We study the spectral theory of bounded-degree Borel pmp graphs, deriving descriptive combinatorial results from the spectral properties of the associated adjacency and Laplacian operators. We first adapt a finite spectral graph theoretic result by Nikiforov to give a new lower bound on the approximate measurable chromatic number of a bounded-degree Borel pmp graph in terms of the maximal spectral values of the adjacency and Laplacian operators. We then derive a broader result on the spectra of bounded self-adjoint operators. Finally, we strengthen a characterization of approximate measurable bipartiteness in terms of the spectrum of an adjacency operator, generalizing it to the case of non-regular graphs.