348 AUDREY TERRAS, ARITHMETICAL QUANTUM CHAOS
of the Kloosterman sums becomes semi-circle in the limit for large p. The top part
of Figure 7 gives the histogram for the spectrum of one of the finite Euclidean plane
graphs. The bottom p art of Figure 7 gives the level spacing for the same Cayley
graph, which does indeed look to be Poisson.
Spectrum for Ep9.1)4020- 80 ·60. 40. 20 20 40 60 80 100
Level Spacing for Ep9,1)4002000.1 0.2 03 04 0.5 0 .6 07 0.8 0.9
Figure 7. The top histogram is the spectrum (without multiplicity) of the
adjacency matrix of the finite Euclidean plane graph X(IF~, S), where p = 1723
and S consists of solutions to the congruence xi + x~ = 1 (mod p). The bottom
is the unnormalized level spacing histogram for the same graph.However when one replaces the 2-dimensional Euclidean graphs with three-
dimensional analogues, the Kloosterman sums can be evaluated as sines or cosines.
If we throw out the half of the eigenvalues of E 48611 (3, 1) that are 0, we obtain the
histogram in the top of Figure 8 which is definitely not the semi-circle distribution.
If we had kept the 0 eigenvalues, there would also be a peak at the origin. The
normalized level spacing histogram for E 48611 (3,1) is in the bottom of Figure 8.
And the bottom does not appear to be Poisson either. See Terras [82] for more
information. Bannai et al [7] have generalized these Euclidean graphs to association
schemes for finite orthogonal groups.
The finite Euclidean graphs can be Ramanujan or not depending on their mood.
For example , in 3 dimensions as in Figure 8, one finds that the graphs are not Ra-
manujan when the prime p = 3(mod 4) is larger than 158. The 3 dimensional graphs
as in Figure 8 are always Ramanujan when p = l(mod4) In the 2-dimensional
case as in Figure 7, one finds that the graphs are always Ramanujan if p = 3(mod4)