Structural properties of graphs of diameter 2 with maximal repeats

It was shown using eigenvalue analysis by Erdös et al. that with the exception of C4C4, there are no graphs of diameter 2, of maximum degree d   and of order d2d2, that is, one less than the Moore bound. These graphs belong to a class of regular graphs of diameter 2, and having certain interesting structural properties, which will be proved in this paper.