Approaching the Moore bound for diameter two by Cayley graphs

The order of a graph of maximum degree d and diameter 2 cannot exceed d2+1, the Moore bound for diameter two. A combination of known results guarantees the existence of regular graphs of degree d, diameter 2, and order at least d2−2d1.525 for all sufficiently large d, asymptotically approaching the Moore bound. The corresponding graphs, however, tend to have a fairly small or trivial automorphism group and the nature of their construction does not appear to allow for modifications that would result in a higher level of symmetry. The best currently available construction of vertex-transitive graphs of diameter 2 and preassigned degree gives order for all degrees of the form d=(3q−1)/2 for prime powers .In this note we show that for an infinite set of degrees d there exist Cayley, and hence vertex-transitive, graphs of degree d, diameter 2, and order d2−O(d3/2).