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

For a graph G and a positive integer c, let Mc(G) be the size of a subgraph of G induced by a randomly sampled subset of c vertices. Second-order moments of Mc(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.