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

For a graph $G$ and a positive integer $c$, let $M_c(G)$ be the size of a subgraph 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 use this fact, coupled to classical moment inequalities, to prove graph theoretical results, to give combinatorial identities, to bound the size of the $c$-densest subgraph from below and the size of the $c$-sparsest subgraph from above, and to provide bounds for approximate enumeration of trivial subgraphs.