Spectra of hypergraphs and applications

To a regular hypergraph we attach an operator, called its adjacency matrix, and study the second largest eigenvalue as well as the overall distribution of the spectrum of this operator. Our definition and results extend naturally what is known for graphs, including the analogous threshold bound 2 √k - 1 for k-regular graphs. As an application of our results, we obtain asymptotic behavior, as N tends to infinity, of the dimension of the space generated by classical cusp forms of weight 2 level N and trivial character which are eigenfunctions of a fixed Hecke operator T_p with integral eigenvalues.