On Weighted Monotone and Subadditive Graphs

TL;DR

This paper explores how to construct the largest monotone and subadditive functions that are minorants of any given weighted graph function, providing a theoretical foundation for such graph properties.

Contribution

It introduces methods to construct the greatest monotone and subadditive minorants for any weighted graph function, expanding understanding of graph weight functions.

Findings

Existence of largest monotone minorant for any weighted graph

Existence of largest subadditive minorant for any weighted graph

Framework for constructing these minorants systematically

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 $H\subseteq G$ .