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It is known that triangle-free graphs of diameter \(2\) are just maximal triangle-free graphs. Kantor ([5]) showed that if \(G\) is a triangle-free and \(4\)-cycle free graph of diameter \(2\), then \(G\) is either a star or a Moore graph of diameter \(2\); if \(G\) is a \(4\)-cycle free graph of diameter \(2\) with at least one triangle, then \(G\) is either a star-like graph or a polarity graph (defined from a finite projective plane with polarities) of order \(r^2 + r + 1\) for some positive integer \(r\) (or \(P_r\)-\emph{graph} for short). We study, by purely graph theoretical means, the structure of \(P_r\)-graphs and construct \(P_r\)-graphs for small values of \(r\). Further, we characterize graphs of diameter \(2\) without \(5\)-cycles and \(6\)-cycles, respectively. In general, one can characterize \(C_k\)-free graphs of diameter \(2\) with \(k > 6\) with a similar approach.