Finite analogues of non-Euclidean spaces and Ramanujan graphs

Abstract

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 plane graphs of valency q+1 over the finite field F q are all Ramanujan graphs. In this paper, we obtain further examples of families of Ramanujan graphs, by using previous works on association schemes and the calculations of their character tables, which are in some sense analogues of the finite upper half planes over finite fields, i.e., finite versions of non-Euclidean spaces. A key observation is that in many (but not all) cases, we can obtain a sharper esitimate |θ|≤2 q−2 on eigenvalues, instead of the original |θ|≤2 q , which was proved by Katz. We combine this observation with the ideas of controlling association schemes and the Ennola type dualities, in our previous papers such as Bannai–Hao–Song (J. Combin. Theory Ser. A 54 (1990) 164), Bannai–Hao–Song–Wei (J. Algebra 144 (1991) 189), Bannai–Kwok–Song (Mem. Fac. Sci. Kyushu Univ. Ser. A 44 (1990) 129), Kwok (Graphs Combin. 7 (1991) 39), Tanaka (Master Thesis (2001)), Tanaka (European J. Combin. 23 (2002) 121) and many others. At the end, we remark that for each fixed valency k≥3 there are only finitely many distance-regular Ramanujan graphs of valency k.