$L^p$ Expander Graphs

We discuss how graph expansion is related to the behavior of $L^p$-functions on the covering tree. We show that the non-trivial eigenvalues of the adjacency operator on aa $(q+1)$-regular graph are bounded by $q^{1\mathrm{/}p}+q^{(p-1)\mathrm{/}p}$ - the $L^p$-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 $L^{p+ϵ}$ for every $ϵ>0$. We generalize the result to operators on edges and to bipartite graphs. The work is based on a combinatorial interpretation of representation-theoretic ideas.