标题： 关于加权单调和次可加图 显示英文标题

标题： On Weighted Monotone and Subadditive Graphs

摘要： 设$G(V,E)$为一个图，$\mathscr{H}:=\big\{H:H\subseteq G\big\}$表示$G$的所有可能子图的集合。 然后对于每个非负函数$w:\mathscr{H}\to\mathbb{R_+}$，图$G(V,E,w)$被称为加权图。 一个加权图$G(V,E,w)$被称为单调（递增）的，如果对于任何$H_1,H_2\subseteq G$满足$H_1\subset H_2$，以下不等式成立： $$w\big(H_1\big)\leq w\big(H_2\big). $$ 另一方面，一个加权图$G(V,E,{w})$被称为次可加的，如果对于任何$H_1,H_2\subseteq G$，以下离散泛函不等式成立： $$ {w}\big(H_1\cup H_2\big)\leq {w}\big(H_1\big)+ {w}\big(H_2\big). $$ 我们的主要结果表明，对于任何图$G(V,E,w)$，都可以构造出最大的单调下界和最大的次可加下界。 换句话说，可以制定最大的递增函数$\overline{w}:\mathscr{H}\to\mathbb{R_+}$和次可加函数$\widetilde{w}:\mathscr{H}\to\mathbb{R_+}$，使得$\overline{w}(H)\leq w(H)$和$\widetilde{w}(H)\leq w(H)$分别对所有$H\subseteq G$成立。 显示英文摘要

摘要： Let $G(V,E)$ be a graph, and $\mathscr{H}:=\big\{H:H\subseteq G\big\}$ denote the collection of all possible subgraphs of $G$. Then for each non-negative function $w:\mathscr{H}\to\mathbb{R_+}$, the graph $G(V,E,w)$ is said to be a weighted graph. A weighted graph $G(V,E,w)$ is called monotone (increasing), if for any $H_1,H_2\subseteq G$ with $H_1\subset H_2$, the following inequality holds: $$w\big(H_1\big)\leq w\big(H_2\big). $$ On the other hand, a weighted graph $G(V,E,{w})$ is termed subadditive, if for any $H_1,H_2\subseteq G$, the following discrete functional inequality is satisfied: $$ {w}\big(H_1\cup H_2\big)\leq {w}\big(H_1\big)+ {w}\big(H_2\big). $$ Our main result demonstrates that for any graph $G(V,E,w)$, it is possible to construct both the largest monotone and the greatest subadditive minorants. In other words, it is feasible to formulate the largest increasing function $\overline{w}:\mathscr{H}\to\mathbb{R_+}$ and subadditive function $\widetilde{w}:\mathscr{H}\to\mathbb{R_+}$ such that $\overline{w}(H)\leq w(H)$ and $\widetilde{w}(H)\leq w(H)$ hold respectively for all $H\subseteq G$ .