Finite Fields and Ramanujan Graphs

Finite analogues of non-Euclidean spaces and Ramanujan graphs

This is a companion paper of “Finite Euclidean graphs and Ramanujan graphs” by the same authors. Finite analogues of the Poincaré upper half plane, i.e., finite upper half plane graphs, were studied by many researchers, including Terras, Evans etc. Finally, it was proved by combining works of A. Weil, P. Deligne, R. Evans, H. Stark, N. Katz, W. Li and many others, that the finite upper half pla...

Ramanujan graphs

Ramanujan Graphs

In the last two decades, the theory of Ramanujan graphs has gained prominence primarily for two reasons. First, from a practical viewpoint, these graphs resolve an extremal problem in communication network theory (see for example [2]). Second, from a more aesthetic viewpoint, they fuse diverse branches of pure mathematics, namely, number theory, representation theory and algebraic geometry. The...

Finite Euclidean graphs over ℤ2r are non-Ramanujan

Graphs are attached to 2r where 2r is the ring with 2 r elements using an analogue of Euclidean distance. M.R. DeDeo (2003) showed that these graphs are non-Ramanujan for r 4. In this paper, we will show that finite Euclidean graphs attached to 2r are non Ramanujan for r 2 except for r = 2 and d = 2, 3. Together with the results in Medrano et al. (1998), this implies that finite Euclidean graph...

14.10 Ramanujan and expander graphs

Case f(x) g(x) 1.1 −1 + bxm/2 + xm 1− bxm/2 + x3m 1.2 1 + bxm/2 + xm b+ xm/2 + x5m/2 1.3 −1 + bxm/2 + xm b− bxm + x5m/2 1.4 −1 + bxm/2 + xm −b− x3m/2 + x5m/2 1.5 1 + bxm/2 + xm b+ bx4m/2 + x5m/2 1.6 1 + bxm/2 + xm 1 + xm + x2m 1.7 −1 + bxm/2 + xm b+ xm + x3m/2 1.8 −1 + bxm/2 + xm −b− bxm + x3m/2 1.9 a− xm/3 + xm −a− xm/3 + x3m 1.10 a− xm/3 + xm 1 + x2m/3 + x8m/3 1.11 a+ xm/3 + xm a+ ax2m/3 + x7...