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Diameter of “almost” always connected Erdős-Renyi graphs

Let $G=(V,E)$ be a random Erdős-Renyi Graph, i.e., $G\in\mathcal{G}(n,p)$. It is well known that if $p=(\log n +c +o(1))/n$ with $c\in\Re$ then $$ P(G \text{ is connected})=e^{-e^{-c}}\ . $$ However, ...