Lp-Expander Graphs

We discuss how graph expansion is related to the behavior of Lp-functions on the covering tree. We show that the non-trivial eigenvalues of the adjacency operator on a (q + 1)-regular graph are bounded by q1/p + q(p-1)/p-the Lp-norm of the operator on the covering tree—if and only if properly averaged lifts of functions from the graph to the tree lie in Lp+ε for every ε > 0. We generalize the result to operators on edges and to bipartite graphs.