A Note on Large Graphs of Diameter Two and Given Maximum Degree

Letvt(d, 2) be the largest order of a vertex-transitive graph of degreedand diameter 2. It is known thatvt(d, 2)=d2+1 ford=1, 2, 3, and 7; for the remaining values ofdwe havevt(d, 2)⩽d2−1. The only knowngenerallower bound onvt(d, 2), valid forall d, seems to bevt(d, 2)⩾⌊(d+2)/2⌋ ⌈(d+2)/2⌉. Using voltage graphs, we construct a family of vertex-transitive non-Cayley graphs which shows thatvt(d, 2)⩾(8/9)(d+12)2for alldof the formd=(3q−1)/2, whereqis a prime power congruent with 1 (mod 4). The construction generalizes to all prime powers and yields large highly symmetric graphs for other degrees as well. In particular, ford=7 we obtain as a special case the Hoffman–Singleton graph, and ford=11 andd=13 we have new largest graphs of diameter 2, and degreedon 98 and 162 vertices, respectively.