Numerical Investigation of the Spectrum for Certain Families of Cayley Graphs

In this paper we extend some earlier computations [8]. In particular, the expanding behavior of Cayley graphs of PSL2 (F107) is compared with that of the Cayley graphs for the group A10 . These computations support the (up to now) unvoiced conjecture of Lubotzky that the symmetric groups and projective linear groups have asymptotically different average expanding behavior. We also give a thorough spectral analysis for a natural family of Cayley graphs which does not admit analysis by Selberg's theorem. 1 Introduction Spectral analysis and operator theory have provided some of the main tools for the recent advances in constructions of expander graphs. In particular, by exploiting the various relationships between the second largest eigenvalue of the Laplacian and the expansion coefficient of graphs, families of expanders have been constructed and analyzed. When the graphs of interest are Cayley graphs, techniques from Fourier analysis are especially useful in this analysis. In this pap..