The measurable Kesten theorem

Abstract

We give an explicit bound on the spectral radius in terms of the densities of short cycles in finite d-regular graphs. It follows that the a finite d-regular Ramanujan graph G contains a negligible number of cycles of size less than c log log |G|.We prove that infinite d-regular Ramanujan unimodular random graphs are trees. Through Benjamini-Schramm convergence this leads to the following rigidity result. If most eigenvalues of a d-regular finite graph G fall in the Alon-Boppana region, then the eigenvalue distribution of G is close to the spectral measure of the d-regular tree. In particular, G contains few short cycles.In contrast, we show that d-regular unimodular random graphs with maximal growth are not necessarily trees.

Line-Graph Lattices: Euclidean and Non-Euclidean Flat Bands, and Implementations in Circuit Quantum Electrodynamics

“…Non-bipartite cubic Ramanujan graphs are I = [−3, −2 √ 2) ∪ (2 √ 2, 3]-gapped, and the recent result in Ref. [31] shows that this I is globally maximal. In Appendix E we extend our analysis to give further examples of locally and globally maximal gap intervals.…”

Ramanujan graphings and correlation decay in local algorithms

“…Using the previous lemma we can create probability distributions of R-valued functions on the vertices of T d in the following way. A similar construction was used in [2], see Lemmas 35 and 36. Let f : X → R be a function in L 2 (X).…”

On the local geometry of graphs in terms of their spectra

“…It is a fairly wellunderstood theme that large, d-regular Ramanujan graphs locally resemble the dregular tree in that they contain few short cycles. For an illustration of such results, see [1,4,9,11,12] and references therein. This relation is not as well understood for sparse, irregular graphs.…”