L^p Norms of Eigenfunctions on Regular Graphs and on the Sphere

We prove upper bounds on the L^p norms of eigenfunctions of the discrete Laplacian on regular graphs. We then apply these ideas to study the L^p norms of joint eigenfunctions of the Laplacian and an averaging operator over a finite collection of algebraic rotations of the two-sphere. Under mild conditions, such joint eigenfunctions are shown to satisfy for large p the same bounds as those known for Laplace eigenfunctions on a surface of non-positive curvature.