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.

翻译：对于图$G$和正整数$c$，令$M_c(G)$表示由随机采样的$c$个顶点诱导的子图的大小。$M_c(G)$的二阶矩编码了图$G$的部分结构信息。我们利用这一事实，结合经典矩不等式，证明了图论结果，给出了组合恒等式，从下方界定了$c$-最密子图的大小，从上方界定了$c$-最疏子图的大小，并提供了平凡子图近似枚举的界。