Level Spacings for Cayley Graphs
We investigate the eigenvalue spacing distributions for randomly generated 4-regular Cayley graphs on SL 2(F P ), S 10 and large cyclic groups by numerically calculating their spectra. We present strong evidence that the distributions are Poisson and hence do not follow the Gaussian orthogonal ensemble. Among the Cayley graphs of SL 2(F P ) that we consider are the new expander graphs recently discovered by Y. Shalom. In addition, we use a Markov chain method to generate random 4-regular graphs, and observe that the average eigenvalue spacings are closely approximated by the Wigner surmise.