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For a Rényi–Erdős graph $G(n, p)$, what can we say about the size of the min-cut (in the whole graph)? I'm looking for something like this:

$$ \Pr(\text{min-cut-size} > x) \geq \cdots $$ or $$ \Pr(\text{min-cut-size} = x) \geq \cdots $$ for an $x \in (0, n)$.

Note: I'm NOT looking for asymptotic expressions (i.e. when $n$ is big); in other words, need something that works for small $n$ as well.

bof
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Daniel
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  • If you mean "min-cut bipartition" (into two similar sizes), it's about $n^2p/4$, fairly tightly. (see https://simons.berkeley.edu/sites/default/files/docs/4814/dembo.pdf). If you actually mean the min-cut, then you need to have a relatively high degree before the graph is even connected. Wiki says you need $p \ge \log n/n$ to get it connected. I expected that a similar derivation would tell you the chance of being only 1-connected. Unless you have average degrees growing faster than $O(\log^c n)$, I think your min-cuts are usually $O(1)$. – Alex Meiburg Oct 12 '18 at 03:58

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