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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2510.18884 |
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| _version_ | 1866912663568646144 |
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| author | Goswami, A. R. |
| author_facet | Goswami, A. R. |
| contents | 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 $H\subseteq G$ . |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_18884 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On Weighted Monotone and Subadditive Graphs Goswami, A. R. General Mathematics Primary: 05C22, Secondary: 39B62 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 $H\subseteq G$ . |
| title | On Weighted Monotone and Subadditive Graphs |
| topic | General Mathematics Primary: 05C22, Secondary: 39B62 |
| url | https://arxiv.org/abs/2510.18884 |