Maximum degree in graphs of diameter 2

The purpose of this paper is to prove that, with the exception of C 4 , there are no graphs of diameter 2 and maximum degree d with d 2 vertices . On one hand our paper is an extension of [4] where it was proved that there are at most four Moore graphs of diameter 2 (i .e . graphs of diameter 2, maximum degree d, and d2 + 1 vertices) . We also use the eigenvalue method developed in that paper . On the other hand, our problem originated in [2] . The domination number of a graph G is the smallest integer k such that G has a k-element subset, S, for which every vertex of G either belongs to S or is adjacent to a vertex of S . Authors of [2] constructed a number of graphs of diameter 2 which contained no three of four-cycles and for which the domination number was arbitrarily large . As a rule, the only lower bounds for the domination numbers were obtained from upper bounds on the maximum degree . This suggested the following question : How small may the maximum degree be compared to the number of vertices in graphs of diameter 2? Since a graph of a diameter 2 and maximum degree d may have at most d 2 + 1 vertices, the question can be formulated as follows : given non-negative numbers d and S, is there a graph of diameter 2 and maximum degree d with d 2 + 1 5 vertices? It was proved in [4] that if 5 = 0 then there are graphs corresponding to d = l, 3, and 7 and that, moreover, only one more case, namely of d = 57, is possible . The case 6 = 1 is solved in the next section, and the last section contains some comments concerning the case 5 > l .