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Keywords:
augmented Sombor index; uncicyclic graph; tree; chemical graph; QSPR analysis
augmented Sombor index; uncicyclic graph; tree; chemical graph; QSPR analysis
Summary:
Topological indices are key quantitative descriptors in mathematical chemistry, unchanged under symmetry operations and retaining graph connectivity; they capture molecular structural features to provide insights into molecular stability and chemical properties, becoming indispensable in cheminformatics and theoretical chemistry. Among degree-based indices, the Sombor index is widely concerned for capturing structural information, and motivated by enhanced structural discrimination, the augmented Sombor index (ASO) is defined for a connected graph $\Omega $ with $|V(\Omega )|\geq 3$ as $$ {\rm ASO} (\Omega ) = \sum _{v_iv_j\in E(\Omega )} \sqrt {\frac {{d_i^2 + d_j^2}}{d_i + d_j - 2}}, $$ where $d_i$ and $d_j$ are the degrees of vertices $v_i$ and $v_j$, respectively. Within the scope of this study, we first establish several sharp bounds for the augmented Sombor index and characterize the extremal graphs attaining these bounds. In particular, we determine the minimum value of the ASO index for unicyclic graphs with a prescribed girth and characterize all graphs achieving this minimum. We also identify the second maximum ASO value among trees and characterize the corresponding extremal tree structures. Furthermore, the minimum and maximum values of the ASO index for bipartite graphs and chemical graphs are obtained, together with a complete characterization of the associated extremal graphs. In addition, we characterize the chemical trees that maximize the ASO index. The chemical applicability of the ASO index is investigated through quantitative structure-property relationship (QSPR) analysis, supported by a comparative assessment of several variants of the Sombor index. Finally, we present concluding remarks and outline potential directions for future research on the augmented Sombor index of graphs.
References:
[1] Ahmad, S., Das, K. C.: General Sombor index: A study of branching in trees and solution for maximal trees with prescribed maximum degree. J. Comb. Optim. 50 (2025), Article ID 21, 23 pages. DOI 10.1007/s10878-025-01343-x | MR 4956695 | Zbl 1576.05040
[2] Ahmad, S., Das, K. C., Farooq, R.: On elliptic Sombor index with applications. Bull. Malays. Math. Sci. Soc. (2) 48 (2025), Article ID 108, 30 pages. DOI 10.1007/s40840-025-01894-6 | MR 4910729 | Zbl 1569.05060
[3] Ahmad, S., Farooq, R., Das, K. C.: The general Sombor index of extremal trees with a given maximum degree. MATCH Commun. Math. Comput. Chem. 94 (2025), 825-853. DOI 10.46793/match94-3.21724 | MR 4958197 | Zbl 08091610
[4] Ali, A., Bhatti, A. A., Raza, Z.: Further inequalities between vertex-degree-based topological indices. Int. J. Appl. Comput. Math. 3 (2017), 1921-1930. DOI 10.1007/s40819-016-0213-4 | MR 3680680 | Zbl 1397.05038
[5] Ali, A., Das, K. C., Alanazi, A. M., Tomescu, I.: On diameter-constrained unicyclic graphs and BID indices. Mediterr. J. Math. 23 (2026), Article ID 41, 22 pages. DOI 10.1007/s00009-025-03027-6 | MR 5014261 | Zbl 08168377
[6] Ali, A., Raza, Z., Bhatti, A. A.: Extremal pentagonal chains with respect to bond incident degree indices. Can. J. Chem. 94 (2016), 870-876. DOI 10.1139/cjc-2016-0308
[7] Borovićanin, B., Das, K. C., Furtula, B., Gutman, I.: Bounds for Zagreb indices. MATCH Commun. Math. Comput. Chem. 78 (2017), 17-100. MR 3677416 | Zbl 1471.92406
[8] Das, K. C.: Open problems on Sombor index of unicyclic and bicyclic graphs. Appl. Math. Comput. 473 (2024), Article ID 128647, 10 pages. DOI 10.1016/j.amc.2024.128647 | MR 4718231 | Zbl 1545.05043
[9] Das, K. C.: On the vertex degree function of graphs. Comput. Appl. Math. 44 (2025), Article ID 183, 18 pages. DOI 10.1007/s40314-025-03161-7 | MR 4878285 | Zbl 1574.05045
[10] Das, K. C., Alharbi, M., Bera, J.: Exponential harmonic index and its applications in structure property modeling. Int. J. Quantum Chem. 125 (2025), Article ID e70099, 19 pages. DOI 10.1002/qua.70099
[11] Das, K. C., Ali, A.: On the augmented Sombor index of graphs. MATCH Commun. Math. Comput. Chem. 96 (2026), 587-610. DOI 10.46793/match.96-2.30625 | MR 5034845 | Zbl 08171799
[12] Das, K. C., Bera, J.: Resolving open problems on the Euler Sombor index. MATCH Commun. Math. Comput. Chem. 95 (2026), 901-918. DOI 10.46793/match.95-3.16625 | MR 5016972 | Zbl 08139968
[13] Das, K. C., Gutman, I.: On Sombor index of trees. Appl. Math. Comput. 412 (2022), Article ID 126575, 8 pages. DOI 10.1016/j.amc.2021.126575 | MR 4300333 | Zbl 1510.05037
[14] Das, K. C., Gutman, I., Ali, A.: Augmented Sombor index. MATCH Commun. Math. Comput. Chem. 95 (2026), 523-547. DOI 10.46793/match.95-2.20125 | MR 4977079 | Zbl 08118248
[15] Gao, W.: Extremal trees and molecular trees with respect to the Sombor-index-like graph invariants $\mathcal {SO}_5$ and $\mathcal {SO}_6$. Czech. Math. J. 74 (2024), 927-941. DOI 10.21136/CMJ.2024.0221-24 | MR 4804969 | Zbl 07953687
[16] Gutman, I.: Degree-based topological indices. Croat. Chem. Acta 86 (2013), 351-361. DOI 10.5562/cca2294
[17] Gutman, I.: Geometric approach to degree-based topological indices: Sombor indices. MATCH Commun. Math. Comput. Chem. 86 (2021), 11-16. MR 4773882 | Zbl 1474.92154
[18] Gutman, I., Furtula, B., Ãz, M. S.: Geometric approach to vertex degree-based topological indices: Elliptic Sombor index, theory and application. Int. J. Quantum Chem. 124 (2024), Article ID e27346, 10 pages. DOI 10.1002/qua.27346
[19] Horoldagva, B., Xu, C.: On Sombor index of graphs. MATCH Commun. Math. Comput. Chem. 86 (2021), 703-713. MR 4777333 | Zbl 1474.92157
[20] Javed, S., Sattar, A., Javaid, M.: Extremal unicyclic graphs with fixed leaves via Sombor index. J. Disc. Math. Sci. Cryptogr. 28 (2025), 783-795. DOI 10.47974/JDMSC-1988
[21] Li, Y., Liu, H., Zhang, R.: Quasi-tree graphs with the minimal Sombor indices. Czech. Math. J. 72 (2022), 1227-1238. DOI 10.21136/CMJ.2022.0152-22 | MR 4517610 | Zbl 1563.05050
[22] Li, Z., Shi, L., Zhang, L., Ren, H., Li, F.: On Sombor index of uniform hypergraphs. MATCH Commun. Comput. Chem. 94 (2025), 229-246. DOI 10.46793/match.94-1.229L | MR 4941052 | Zbl 1560.92167
[23] Liu, H.: Comparison between Merrifield-Simmons index and some vertex-degree-based topological indices. Comput. Appl. Math. 42 (2023), Article ID 89, 8 pages. DOI 10.1007/s40314-023-02240-x | MR 4551655 | Zbl 1506.05050
[24] Liu, H., Gutman, I., You, L., Huang, Y.: Sombor index: Review of extremal results and bounds. J. Math. Chem. 60 (2022), 771-798. DOI 10.1007/s10910-022-01333-y | MR 4402726 | Zbl 1492.92156
[25] Liu, H., Huang, Y.: Sharp bounds on the symmetric division deg index of graphs and line graphs. Comput. Appl. Math. 42 (2023), Article ID 285, 13 pages. DOI 10.1007/s40314-023-02428-1 | MR 4630457 | Zbl 1538.05059
[26] Lu, Y., Zhou, Q.: On hyper-Zagreb index conditions for Hamiltonicity of graphs. Czech. Math. J. 72 (2022), 653-662. DOI 10.21136/CMJ.2022.0089-21 | MR 4467933 | Zbl 1538.05156
[27] Lučić, B., Trinajstić, N., Zhou, B.: Comparison between the sum-connectivity index and product-connectivity index for benzenoid hydrocarbons. Chem. Phys. Lett. 475 (2009), 146-148. DOI 10.1016/j.cplett.2009.05.022
[28] Maitreyi, V., Elumalai, S., Balachandran, S.: On the extremal general Sombor index of trees with given pendent vertices. MATCH Commun. Comput. Chem. 92 (2024), 225-248. DOI 10.46793/match.92-1.225M | MR 4888853 | Zbl 1544.92237
[29] Masre, M., Vetrík, T.: General degree-eccentricity index of trees. Bull. Malays. Math. Sci. Soc. (2) 44 (2021), 2753-2772. DOI 10.1007/s40840-021-01086-y | MR 4296311 | Zbl 1471.05023
[30] Miftahurrahman, M., Maulana, F.: Sombor index of non-coprime graphs on generalized quaternion groups. J. Indones. Algebra Soc. 1 (2026), 1-7.
[31] Mohammadi, M., Barzegar, H.: Bounds for Sombor index using topological and statistical indices. J. Discr. Math. Appl. 10 (2025), 61-85.
[32] Mondal, S., Das, K. C.: Zagreb connection indices in structure property modelling. J. Appl. Math. Comput. 69 (2023), 3005-3020. DOI 10.1007/s12190-023-01869-5 | MR 4617124 | Zbl 1522.05052
[33] Movahedi, F.: Results on the elliptic Sombor index. RAIRO, Oper. Res. 59 (2025), 2201-2212. DOI 10.1051/ro/2025086 | MR 4945982 | Zbl 08091151
[34] Nithyaa, P., Elumalai, S., Balachandran, S., Masrec, M.: Ordering unicyclic graphs with a fixed girth by Sombor indices. MATCH Commun. Math. Comput. Chem. 92 (2024), 205-224. DOI 10.46793/match.92-1.205N | MR 4888852 | Zbl 1544.92238
[35] Oboudi, M. R.: Sombor index of a graph and of its subgraphs. MATCH Commun. Comput. Chem. 92 (2024), 697-702. DOI 10.46793/match.92-3.697O | MR 4904647 | Zbl 1542.92225
[36] Randić, M.: Quantitative structure-property relationship: Boiling points of planar benzenoids. New J. Chem. 20 (1996), 1001-1009.
[37] Ravindran, B. A.: Sombor index on trees: A review. MATCH Commun. Comput. Chem. 94 (2025), 291-305. DOI 10.46793/match.94-2.291B | MR 4949008 | Zbl 1567.92203
[38] Raza, Z., Akhter, S., Mondal, S., Ghorbani, M.: Maximal tree and unicyclic graph for Euler Sombor index with given diameter. AKCE Int. J. Graphs Comb. 22 (2025), 289-304. DOI 10.1080/09728600.2025.2517077 | MR 5004754 | Zbl 1576.05099
[39] Sekar, S., Balachandran, S., Elumalai, S., Liu, H.: Maximum Euler Sombor index of unicyclic graphs with given diameter. MATCH Commun. Comput. Chem. 95 (2026), 759-776. DOI 10.46793/match.95-3.12225 | MR 4990175 | Zbl 08139960
[40] Shanmukha, M. C., Basavarajappa, N. S., Shilpa, K. C., Usha, A.: Degree-based topological indices on anticancer drugs with QSPR analysis. Heliyon 6 (2020), Article ID e04235, 9 pages. DOI 10.1016/j.heliyon.2020.e04235
[41] Shigehalli, V. S., Chougale, S. A.: Sombor index and its applications. Open J. Disc. Appl. Math. 8 (2025), 50-63. DOI 10.30538/psrp-odam2025.0111 | MR 4896579
[42] Stein, W.: SageMath: Version 6.8. Available at https://www.sagemath.org/ (2015).
[43] Xu, C., Horoldagva, B., Buyantogokh, L.: The exponential second Zagreb index of $(n,m)$-graphs. Mediterr. J. Math. 20 (2023), Article ID 181, 8 pages. DOI 10.1007/s00009-023-02387-1 | MR 4569798 | Zbl 1511.05052
[44] Xu, C., Li, G.: Maximum values of Sombor index of bicyclic graphs with a given matching number. RAIRO, Theor. Inform. Appl. 57 (2023), Article ID 11, 11 pages. DOI 10.1051/ita/2023005 | MR 4674493 | Zbl 1531.05056
[45] Yang, F., Lu, M., Guo, J.: On the minimum Kirchhoff index of unicyclic graphs with given girth and diameter. Bull. Malays. Math. Sci. Soc. (2) 45 (2022), 1287-1299. DOI 10.1007/s40840-022-01246-8 | MR 4412322 | Zbl 1487.05068
[46] Zhang, Q., Tchier, F., Hussain, Z., Aslam, A., Zuo, X.: On the bounds for the Euler Sombor index of molecular trees and its applications. J. Appl. Math. Comput. 71 (2025), S645--S663. DOI 10.1007/s12190-025-02506-z | MR 4956655
[47] Zhang, X., Noureen, S., Kamran, S., Ibrahim, S. M., Aslam, A.: Sombor indices in main group metal chemistry: Computational evaluation of bismuth(III) iodide, oxide/silicate frameworks, and dendrimers for QSAR applications. Main Group Metal Chem. 48 (2025), Article ID 20250011, 11 pages. DOI 10.1515/mgmc-2025-0011
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