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In this paper we consider the problem of characterizing directed graphs of specified diameter. We are especially interested in the minimal number of arcs \(\textbf{a(d,n)}\) required to construct a directed graph on \(n\) vertices with diameter \(d\). Classes of graphs considered include general digraphs, digraphs without cycles of length \(2\), and digraphs with regular indegree or regular outdegree. Upper bounds are developed in cases where the exact solutions are not known.