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Let $G(n,p)$ be an Erdős–Rényi graph on $n$ vertices. Is there an explicit expression for the probability $P_{n,p}(u,v)$ that two fixed (distinct) vertices $u,v$ lie in the same connected component of $G(n,p)$?

I'm familiar with the standard asymptotic results about connected components in Erdős–Rényi graphs but was unable to find explicit results for $P_{n,p}$ for finite $n$. I expect these probabilities to be polynomials in $p$ of degree $n(n-1)/2$ but did not succeed in determining the coefficients for general $n$ and $p$.

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Explicit expressions for the all-terminal reliabilty $R$ were established back in 1959 by Gilbert. The two-terminal reliability function (which is what the question was about) can then be computed in terms of $R$.

See also the corresponding post on mathoverflow.

Gilbert, E. N. Random graphs. Ann. Math. Statist. 30 1959 1141--1144. MR0108839 (21 #7551)

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