For any pair of natural numbers and d, -graph is a graph with maximum degree at most and diameter at most d. In a graph with maximum degree the number of vertices at distances i from any given vertex is at most . Therefore the number of vertices in -graph cannot exceed the Moore bound: . This paper only considers the case and only considers graphs with vertices. Therefore it is sufficient to consider cubic graphs, for if a (3, d)-graph has a vertex of degree at most 2, then the graph has at most vertices. It is well known that for and Moore graphs, i.e., -graphs with vertices, do not exist (see Biggs). Bannai and Ito state that a ( , d)-graph cannot have exactly vertices. For this is trivial, as cubic graphs must have an even number of vertices.
