In 1960, Hoffman and Singleton investigated the existence of Moore graphs of diameter 2 (graphs of maximum degree \(d\) and \(d^2 + 1\) vertices), and found that such graphs exist only for \(d = 2, 3, 7\) and possibly \(57\). In 1980, Erdős et al., using eigenvalue analysis, showed that, with the exception of \(C_4\), there are no graphs of diameter 2, maximum degree \(d\) and \(d^2\) vertices. In this paper, we show that graphs of diameter 2, maximum degree \(d\) and \(d^2 – 1\) vertices do not exist for most values of \(d\) with \(d \geq 6\), and conjecture that they do not exist for any \(d \geq 6\).