On Weighted Monotone and Subadditive Graphs

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 $H\subseteq G$ .