We examine the adjacency matrices of three-regular graphs representing
one-face maps. Numerical studies reveal that the limiting eigenvalue statistics
of these matrices are the same as those of much larger, and more widely studied
classes from Random Matrix Theory. We present an algorithm for generating
matrices corresponding to maps of genus zero, and find the eigenvalue
statistics in the genus zero case differ strikingly from those of higher genus.
These results lead us to conjecture that the eigenvalue statistics depend on
the rigidity of the underlying map, and the distribution of scaled eigenvalue
spacings shifts from that of the Gaussian Orthogonal Ensemble to the
exponential distribution as the map size increases relative to the genus.