Some Large Trivalent Graphs Having Small Diameters

Kantor, W.M., Some large trivalent graphs having small diameters, Discrete Applied Mathematics 37/38 (1992) 353-357. If Iz 2 10, then there is a tnvalent Cayley graph for G =PSL (n. y) whose diameter is O(logla). This paper concerns an improvement of a result of Babai, Kantor and Lubotzky [ 11. In that paper it was shown that there is a constant C such that every non-Abelian finite simple group G has a set S of seven generators for which d(G, S) I C log 1 GI . Here, S was a carefully chosen generating set for G, and d(G,S) denotes the diameter of the corresponding undirected Cayley graph. This bound is best possible, since a simple count (the ‘Yvloore bound”) shows that d(G, S) + 1> log2 is/ (ICI>* In this paper we will decrease ISI so as to have ISI = 2 and ISU S-’ I = 3 in case G = PSL(n, q) with n 2 10: Theorem. If n 2 10, then there is a trivalent (undirected) Cayley graph for G = PSL(n, q) whose diameter is O(log ICI). Moreover, there is an algorithm which, when given g E G, finds a word in S representing g in O(log IGl) steps (i.e., multiplications and inversions of elements of S). Actually, we will onl;r need to assume that n 18 when q is even. There are analogous results obtainable by similar arguments for all the finite simple groups of Lie type, provided that the ranks are not too small. Steinberg [2] obtained two generators for each finite group of Lie type; but his generators do not include an involution, and hr ‘c argument does not produce the desired diameter bound. * This research was supported in part by NSF grant DMS 87-01794 and NSA grant MDA 904-88-H-2040. 0166-218X/92/$05.00