$L^p$ Expander Graphs

Abstract

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 $q^{1/p}+q^{(p-1)/p}$ - 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 $L^{p+\epsilon }$ for every $\epsilon >0$. We generalize the result to operators on edges and to bipartite graphs. The work is based on a combinatorial interpretation of representation-theoretic ideas.

29 pages. Final version. To appear in Israel Journal of Mathematics

References in corpus (3)

• The Ramanujan Property for Simplicial Complexes
• Lp expander Complexes
• Cutoff on Graphs and the Sarnak-Xue Density of Eigenvalues

Cited by in corpus (2)

• Cutoff on Hyperbolic Surfaces
• Cutoff on Graphs and the Sarnak-Xue Density of Eigenvalues