Graph spectra for finite upper half planes over rings

AbstractGraphs attached for finite analogues of the Poincaré upper half plane over rings ZprZ are introduced for p an odd prime. The spectra of these 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 the graphs are not Ramanujan graphs when r = 2 and p ⩾ 5. Histograms of the eigenvalues for r = 1 and 2 are also compared. The graphs over rings are of interest for connections with p-adic upper half planes as well as fundamental domains for congruence subgroups of the modular group SL(2, Z)