GRAPH SPECTRA FOR FINITE UPPER HALF-PLANES OVER RINGS

Graphs attached for finite analogues of the Poincare upper half plane over rings Z/p(r)Z are introduced for p an odd prime. The spectra of t hese graphs for r = 2 are shown to be related to those for r = 1 which were studied earlier. In contrast to the case r = 1 , we find that th e graphs are not Ramanujan graphs when r = 2 and p greater than or equ al to 5. Histograms of the eigenvalues for r = 1 and 2 are also compar ed. The graphs over rings are of interest for connections with p-adic upper half planes as well as fundamental domains for congruence subgro ups of the modular group SL(2, Z).