On the Diminished Sombor Index of Fixed-Order Molecular Graphs With Cyclomatic Number at Least 3
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arXiv
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2025
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| author | Alotaibi, Abdulaziz Mutlaq Alanazi, Abdulaziz M. Hassan, Taher S. Ali, Akbar |
| author_facet | Alotaibi, Abdulaziz Mutlaq Alanazi, Abdulaziz M. Hassan, Taher S. Ali, Akbar |
| contents | For a graph $G$ with edge set $E$, let $d(u)$ denote the degree of a vertex $u$ in $G$. The diminished Sombor (DSO) index of $G$ is defined as $DSO(G)=\sum_{uv\in E}\sqrt{(d(u))^2+(d(v))^2}(d(u)+d(v))^{-1}$. The cyclomatic number of a graph is the smallest number of edges whose removal makes the graph acyclic. A connected graph of maximum degree at most $4$ is known as a molecular graph. The primary motivation of the present study comes from a conjecture concerning the minimum DSO index of fixed-order connected graphs with cyclomatic number $3$, posed in the recent paper [F. Movahedi, I. Gutman, I. Redžepović, B. Furtula, Diminished Sombor index, MATCH Commun. Comput. Chem. 95 (2026) 141--162]. The present paper gives all graphs minimizing the DSO index among all molecular graphs of order $n$ with cyclomatic number $\ell$, provided that $n\ge 2(\ell-1)\ge4$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_12294 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Diminished Sombor Index of Fixed-Order Molecular Graphs With Cyclomatic Number at Least 3 Alotaibi, Abdulaziz Mutlaq Alanazi, Abdulaziz M. Hassan, Taher S. Ali, Akbar Combinatorics For a graph $G$ with edge set $E$, let $d(u)$ denote the degree of a vertex $u$ in $G$. The diminished Sombor (DSO) index of $G$ is defined as $DSO(G)=\sum_{uv\in E}\sqrt{(d(u))^2+(d(v))^2}(d(u)+d(v))^{-1}$. The cyclomatic number of a graph is the smallest number of edges whose removal makes the graph acyclic. A connected graph of maximum degree at most $4$ is known as a molecular graph. The primary motivation of the present study comes from a conjecture concerning the minimum DSO index of fixed-order connected graphs with cyclomatic number $3$, posed in the recent paper [F. Movahedi, I. Gutman, I. Redžepović, B. Furtula, Diminished Sombor index, MATCH Commun. Comput. Chem. 95 (2026) 141--162]. The present paper gives all graphs minimizing the DSO index among all molecular graphs of order $n$ with cyclomatic number $\ell$, provided that $n\ge 2(\ell-1)\ge4$. |
| title | On the Diminished Sombor Index of Fixed-Order Molecular Graphs With Cyclomatic Number at Least 3 |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2509.12294 |