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, I found no information about the expectation of the diameter if $$G$$ is connected.

My question is: Let $$G\in\mathcal{G}(n,p)$$ with $$p=(\log n +c +o(1))/n$$ and $$c\in\Re$$. What is the value of $$\mathbb{E}(diam\ G \mid G\text{ is connected})$$

Is the result known? Or is it an open question? Any reference is welcome.

Motivation:

I'm performing simulations of several algorithms to compute the diameter of random graphs. Some of them are generated using the value of $$p$$ above. A graph is used only if it is connected. I would like to compare the results with the theoretical expectation.

• Setting $c=1$, it appears numerically that $\mathbb{E}(\operatorname{diam} G | G \text{ is connected}) \leq \log n$ for large enough $n$ in an experiment I just ran. (Indeed, it appears to grow much slower than this.) – cdipaolo Jun 25 '19 at 6:13
• @cdipaolo Yes, I performed simulations with different values of $c$ and it seems that it is bounded by $O(\log n)$. I'm interested in a theoretical formula. – Lamine Jun 25 '19 at 14:10