  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 $2$-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.
