ABSTRACT
This chapter presents introduction to the maximum order digraphs for diameter 2 or degree 2. It considers finite digraphs, that is, digraphs in which both V and are finite. The number of vertices in the digraph is called the order of the digraph. The chapter deals with an extension of the Moore graph problem. This problem was originally posed for regular undirected graphs. Thus, apart from the fact that it is not known whether or not a Moore graph with diameter 2 and degree 57 exists, the Moore graph problem for undirected graphs has been solved. The chapter also presents results of the maximum order digraphs for diameter 2 or degree 2 with theorems and proofs as well as lemmas.
