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For a graph $G$ and a positive integer $c$, let $M_c(G)$ be the size of asubgraph of $G$ induced by a randomly sampled subset of $c$ vertices.Second-order moments of $M_c(G)$ encode part of the structure of $G$. We usethis fact, coupled to classical moment inequalities, to prove graph theoreticalresults, to give combinatorial identities, to bound the size of the $c$-densestsubgraph from below and the size of the $c$-sparsest subgraph from above, andto provide bounds for approximate enumeration of trivial subgraphs.

Second-order moments of the size of randomly induced subgraphs of given order