  We give explicit estimates between the spectral radius and the densities of
short cycles for finite d-regular graphs. This allows us to show that the
essential girth of a finite d-regular Ramanujan graph G is at least c log log
|G|.
  We prove that infinite d-regular Ramanujan unimodular random graphs are
trees. Using Benjamini-Schramm convergence this leads to a rigidity result
saying that 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.
  Kesten showed that if a Cayley graph has the same spectral radius as its
universal cover, then it must be a tree. We generalize this to unimodular
random graphs.
