Finite analogues of Euclidean space

Abstract

Graphs are attached to double-struck Fnq, where double-struck Fq is the field with q elements, q odd, using an analogue of the Euclidean distance. The graphs are shown to be asymptotically Ramanujan for large q (better than Ramanujan in half the cases). Comparisons are made with finite upper half planes constructed in a similar way using an analogue of Poincaré's non-Euclidean distance. The eigenvalues of the adjacency operators of the finite Euclidean graphs are shown to be Kloosterman sums.