The well-known degreeÂdiameter problem asks for determining the largest possible number n (d, k) of vertices in a graph of maximum degree d and diameter k. For information about the current state of the general problem we refer to [1, 5 8, 14]; in this note we focus only on the special case of graphs of diameter 2. The equality in n (d, 2) d2+ 1 (where the right-hand side is known as the Moore bound) is attainable only if d= 1, 2, 3, 7 and, possibly, 57; except possibly for the last value, the corresponding extremal graphs K2, C5, the Petersen graph, and the Hoffman Singleton graph are unique [15]. For the remaining d it was shown [10] that n (d, 2) d2&1; so far the only values of d for which this bound is known to be attained are d= 4 and 5 (see [9] for the extremal graphs). The asymptotically best lower bound on n (d, 2) can be obtained as follows. Incidence graphs associated with finite projective planes [3] give the inequality n (d, 2) d2&d+ 1 for each d such that d&1 is a prime power.(The paper [3] appeared a long time ago but somehow did not receive appropriate attention. As observed in [10] and rediscovered in [7], the bound can further be improved to n (d, 2) d2&d+ 2 if d&1 is a power of 2.) For the remaining values of d, one can use the fact that for each d there is a prime between d and d+ c= d 7Â12+=, where=> 0 is arbitrary and c= is a constant; see [16]. Combining this with vertex duplication in the graphs of [3] yields n (d, 2) d2&c $= d19Â12+= for some constant c $= and all d.

Nevertheless, not much is known about the vertex-transitive version of the problem. Let vt (d, 2) be the largest order of a vertex-transitive graph of degree d and diameter 2. Obviously, we have the same upper bounds on vt (d, 2) as on n (d, 2), and the above comments are valid with two exceptions. First, there is no vertex-transitive graph of diameter two and degree 57; see [4]. Second, we do not know if the bound vt (d, 2) d2&1 for d {1, 2, 3, 7 can be attained. As regards lower bounds, unfortunately, the graphs of [3] are not vertex-transitive. Presently, the best general result (that is, valid for all d) seems to be [12]