On Weighted Monotone and Subadditive Graphs
A. R. Goswami
TL;DR
This paper studies how to systematically construct best-possible monotone and subadditive minorants for weights on graph substructures. It introduces explicit formulas for the largest monotone minorant $\overline{w}(H)=\min\{w(H'):\ H\subseteq H'\}$ and the largest subadditive minorant $\widetilde{w}(H)=\min\{w(H_1)+\cdots+w(H_n): \bigcup_i H_i=H\}$, proving they are valid minorants and maximizing properties. It further analyzes how these constructions interact with monotonicity, conditional equivalences for majorants, and corollaries, along with extensions to infinite graphs and practical computational remarks. The results provide a principled way to enclose any given weight function between best-possible monotone/subadditive envelopes with potential applications to graph-analytic inequalities and related optimization problems.
Abstract
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 $H\subseteq G$ .