Diameters of finite upper half plane graphs

Let GF ( q ) be a finite field of q elements. Let G denote the group of matrices M ( x, y ) = ( y x 0 1 ) over GF ( q ) with y ≠ 0. Fix an irreducible polynomial $$u^2+tu+n \in GF(q)[u]\,.$$ For each a ϵ GF ( q ), let X a be the graph whose vertices are the q 2 − q elements of G , with two vertices M ( x, y ), M ( v, w ) joined by an edge if and only if $$(x-v)^2+t(x-v)(y-w)+n(y-w)^2 = w y a\,.$$ The graphs X a with a ϵ/ {0, t 2 − 4 n } are ( q + 1)‐regular connected graphs which have received recent attention, as they've been shown to be Ramanujan graphs. We determine the diameter of these graphs X a . © 1996 John Wiley & Sons, Inc.