On the difference between Sombor index and harmonic-arithmetic index and its applications

Zhenhua Su; Zhenhua Su

doi:10.3934/era.2026265

Electronic Research Archive

2026, Volume 34, Issue 9: 5993-6010. doi: 10.3934/era.2026265

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On the difference between Sombor index and harmonic-arithmetic index and its applications

Zhenhua Su ^{1,2
,
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1.
School of Mathematics and Computational Sciences, Huaihua University, Huaihua, Hunan, China
2.
Key Laboratory of Intelligent Control Technology for Wuling-Mountain Ecological Agriculture in Hunan Province, Huaihua, Hunan, China

Received: 30 April 2026 Revised: 16 June 2026 Accepted: 29 June 2026 Published: 13 July 2026

The Sombor index ($ SO $) and harmonic-arithmetic index ($ HA $) stand as two well-established topological indices constructed based on vertex degrees. The difference between the $ SO $ index and $ HA $ index of a graph $ G $ is introduced as
$ (SO-HA)(G) = \sum\limits_{{uv\in E(G)}}\sqrt{d(u)^2+d(v)^2}-\sum\limits_{{uv\in E(G)}}\frac{4d(u)d(v)}{(d(u)+d(v))^2}, $
where $ d(u) $ stands for the degree of vertex $ u\in V(G) $. In the present work, we determined the trees, unicyclic graphs, and bicyclic graphs that achieved extremal $ SO-HA $ values, characterized the chemical graphs with extremal $ SO-HA $ values, and obtained the trees with fixed maximum degree minimizing the $ SO-HA $ index. Furthermore, by performing QSPR analysis on octane isomers, we observed that the $ SO-HA $ index possessed favorable physicochemical predictability.
- Sombor index,
- harmonic-arithmetic index,
- difference,
- QSPR analysis
Citation: Zhenhua Su. On the difference between Sombor index and harmonic-arithmetic index and its applications[J]. Electronic Research Archive, 2026, 34(9): 5993-6010. doi: 10.3934/era.2026265

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Abstract

The Sombor index ($ SO $) and harmonic-arithmetic index ($ HA $) stand as two well-established topological indices constructed based on vertex degrees. The difference between the $ SO $ index and $ HA $ index of a graph $ G $ is introduced as

$ (SO-HA)(G) = \sum\limits_{{uv\in E(G)}}\sqrt{d(u)^2+d(v)^2}-\sum\limits_{{uv\in E(G)}}\frac{4d(u)d(v)}{(d(u)+d(v))^2}, $

where $ d(u) $ stands for the degree of vertex $ u\in V(G) $. In the present work, we determined the trees, unicyclic graphs, and bicyclic graphs that achieved extremal $ SO-HA $ values, characterized the chemical graphs with extremal $ SO-HA $ values, and obtained the trees with fixed maximum degree minimizing the $ SO-HA $ index. Furthermore, by performing QSPR analysis on octane isomers, we observed that the $ SO-HA $ index possessed favorable physicochemical predictability.

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