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 $c\log\log\vert G\vert$.
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.
"The measurable Kesten theorem." Ann. Probab. 44 (3) 1601 - 1646, May 2016. https://doi.org/10.1214/14-AOP937