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An Erdos-Renyi graph over $n$ vertices and average degree $d$ is not connected whp iff $d < \log n$. I was wondering for what the degree $d$ would a random regular graph of degree $d$ be connected? In particular will this be connected for some degree less than $\log n$?

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  • $\begingroup$ BTW, Vivek, you should accept answers if they do answer the question as posed. $\endgroup$ – Sasho Nikolov Mar 2 '17 at 15:15
  • $\begingroup$ I will to read/glance the paper and then accept the answer. Thanks for letting me know though! $\endgroup$ – Vivek Bagaria Mar 2 '17 at 20:40
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For constant $d \geq 3$, a random $d$-regular graph is connected with high probability. In fact, it is an expander with high probability. See for example this note by David Ellis. Friedman even showed that a random $d$-regular graph has nearly optimal spectral gap, with high probability.

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