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

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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

$(S O-H A)(G)=\sum_{u v \in E(G)} \sqrt{d(u)^{2}+d(v)^{2}}-\sum_{u v \in E(G)} \frac{4 d(u) d(v)}{(d(u)+d(v))^{2}},$

where d(u) stands for the degree of vertex u ∈ 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.

Keywords

Sombor index / harmonic-arithmetic index / difference / QSPR analysis

CHANGE HISTORY

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References

[1] J. A. Bondy, U. S. R. Murty, Graph theory with applications, Elsevier, New York, 1976.

[2] H. Wiener, Structural determination of paraffin boiling points, J. Amer. Chem. Soc., 69 (1947), 17-20.

[3] I. Gutman, Geometric approach to degree-based topological indices: Sombor indices, MATCH Commun. Math. Comput. Chem., 86 (2021), 11-16.

[4] R. Cruz, J. Rada, Extremal values of the Sombor index in unicyclic and bicyclic graphs, J. Math. Chem., 59 (2021), 1098-1116. https://doi.org/10.1007/s10910-021-01232-8

[5] K. C. Das, Open problems on Sombor index of unicyclic and bicyclic graphs, Appl. Math. Comput., 473 (2024), 128647. https://doi.org/10.1016/j.amc.2024.128647

[6] I. Redžepović, Chemical applicability of Sombor indices, J. Serb. Chem. Soc., 86 (2021), 445-457. https://doi.org/10.2298/JSC201215006R

[7] H. Liu, L. You, Y. Huang, Ordering chemical graphs by Sombor indices and its applications, MATCH Commun. Math. Comput. Chem., 87 (2022), 5-22. https://doi.org/10.46793/match.87-1.005L

[8] S. Filipovski, Relations between Sombor index and some degree-based topological indices, Iran. J. Math. Chem., 12 (2021), 19-26. https://doi.org/10.22052/IJMC.2021.240385.1533

[9] Z. Wang, Y. Mao, Y. Li, B. Furtula, On relations between Sombor and other degree-based indices, J. Appl. Math. Comput., 68 (2022), 1-17. https://doi.org/10.1007/s12190-021-01516-x

[10] H. Liu, I. Gutman, L. You, Y. Huang, Sombor index: Review of extremal results and bounds, J. Math. Chem., 66 (2022), 771-798. https://doi.org/10.1007/s10910-022-01333-y

[11] B. A. Ravindrana, Sombor index on trees-A review, MATCH Commun. Math. Comput. Chem., 94 (2025), 291-305. https://doi.org/10.46793/match.94-2.291B

[12] A. M. Albalahi, A. Ali, A. M. Alanazi, A. A. Bhatti, A. E. Hamza, Harmonic-arithmetic index of (molecular) trees, preprint, arXiv:2302.11099.

[13] K. Ramesh, S. Loganathan, Harmonic-Arithmetic index: Trees with maximum degrees and comparative analysis of antidrugs, Symmetry, 17 (2025), 167. https://doi.org/10.3390/sym17020167

[14] G. Ding, L. Zhong, X. Wang, Harmonic-Arithmetic index of unicyclic graphs, Iran J. Math. Chem., 16 (2025), 93-108. https://doi.org/10.22052/IJMC.2024.255486.1909

[15] R. Kalpana, L. Shobana, Harmonic-Arithmetic index: Lower bound for n-order trees with fixed number of pendant vertices and monogenic semigroup for graph operations , Int. J. Math. Math. Sci., 2025 (2025), 5576652. https://doi.org/10.1155/ijmm/5576652

[16] A. Ali, E. Milovanović, S. Stankov, M. Matejić, I. Milovanović, Inequalities involving the harmonic-arithmetic index, Afr. Mat., 35 (2024), 46. https://doi.org/10.1007/s13370-024-01183-8

[17] K. Aarthi, S. Elumalai, S. Balachandran, S. Mondal, On difference between atom-bond sum-connectivity index and Randić index of graphs, J. Appl. Math. Comput., 71 (2025), 2727-2748. https://doi.org/10.1007/s12190-024-02339-2

[18] Z. Su, H. Deng, A note on the extremal graphs with respect to Vertex-degree-based topological indices for c-cyclic graphs, MATCH Commun. Math. Comput. Chem., 95 (2026), 163-178. https://doi.org/10.46793/match.95-1.12725