The higher-order Sombor index

Document Type : Original paper


1 Hunan Normal University

2 College of Mathematics and Computer Science, Hunan Normal University, Changsha, Hunan 410081, P. R. China


Based on the geometric background of Sombor index and motivating by the higher order connectivity index and the Sombor index, we introduce the pathcoordinate of a path in a graph and a degree-point in a higher dimensional coordinate system, and define the higher order Sombor index of a graph. We first consider mathematical properties of the higher order Sombor index, show that the higher order connectivity index of a starlike tree is completely determined by its branches and that starlike trees with a given maximum degree which have the same higher order Sombor indices are isomorphic. Then, we determine the extremal values of the second order Sombor index for all trees with n vertices and characterize the corresponding extremal trees. Finally, the chemical importance of the second order Sombor index is investigated and it is shown that the new index is useful in predicting physicochemical properties with high accuracy compared to some well-established.


Main Subjects

[1] S. Alikhani and N. Ghanbari, Sombor index of polymers, MATCH Commun. Math. Comput. Chem. 86 (2021), no. 3, 715–728.
[2] S. Amin, A.U. Rehman Virk, M.A. Rehman, and N.A. Shah, Analysis of dendrimer generation by Sombor indices, J. Chem. 2021 (2021), Article ID. 9930645.
[3] O. Araujo and J.A. De La Pena, The connectivity index of a weighted graph, Lin. Alg. Appl. 283 (1998), no. 1-3, 171–177.–3795(98)10096–4
[4] H. Chen, W. Li, and J. Wang, Extremal values on the Sombor index of trees, MATCH Commun. Math. Comput. Chem. 87 (2022), no. 1, 23–49.
[5] R. Cruz, I. Gutman, and J. Rada, Sombor index of chemical graphs, Appl. Math. Comput. 399 (2021), Article ID: 126018.
[6] K.C. Das, A.S. Çevik, I.N. Cangul, and Y. Shang, On Sombor index, Symmetry 13 (2021), no. 1, Article  ID: 140
[7] H. Deng, Z. Tang, and R. Wu, Molecular trees with extremal values of Sombor indices, Int. J. Quantum Chem. 121 (2021), no. 11, Article ID: e26622.
[8] X. Fang, L. You, and H. Liu, The expected values of Sombor indices in random hexagonal chains, phenylene chains and Sombor indices of some chemical graphs, Int. J. Quantum Chem. 121 (2021), no. 17, Article ID: e26740.
[9] I. Gutman, Geometric approach to degree-based topological indices: Sombor indices, MATCH Commun. Math. Comput. Chem. 86 (2021), no. 1, 11–16.
[10] I. Gutman, Some basic properties of Sombor indices, Open J. Discrete Appl. Math. 4 (2021), no. 1, 1–3.–odam2021.0047
[11] I. Gutman, Sombor indices–back to geometry, Open J. Discrete Appl. Math. 5 (2022), no. 2, 1–5.–odam2022.0072
[12] I. Gutman, TEMO theorem for Sombor index, Open J. Discrete Appl. Math. 5 (2022), no. 1, 25–28.–odam2022.0067
[13] I. Gutman, N. Gürsoy, A. Gürsoy, and A. Ülker, New bounds on Sombor index, Commun. Comb. Optim. 8 (2023), no. 2, 305–311.
[14] B. Horoldagva and C. Xu, On Sombor index of graphs, MATCH Commun. Math. Comput. Chem. 86 (2021), no. 3, 703–713.
[15] L.B. Kier, W.J. Murray, M. Randić, and L.H. Hall, Molecular connectivity V: connectivity series concept applied to density, J. Pharm. Sci. 65 (1976), no. 8, 1226–1230.
[16] H. Liu, H. Chen, Q. Xiao, X. Fang, and Z. Tang, More on Sombor indices of chemical graphs and their applications to the boiling point of benzenoid hydrocarbons, Int. J. Quantum Chem. 121 (2021), no. 17, Article ID: e26689.
[17] J. Rada and O. Araujo, Higher order connectivity index of starlike trees, Discrete Appl. Math. 119 (2002), no. 3, 287–295.–218X(01)00232–3
[18] J. Rada, J. Rodríguez, and J.M. Sigarreta, General properties on Sombor indices, Discrete Appl. Math. 299 (2021), 87–97.
[19] M. Randić, Characterization of molecular branching, J. Amer. Chem. Soc. 97 (1975), no. 23, 6609–6615.
[20] I. Redžepović, Chemical applicability of Sombor indices, J. Serb. Chem. Soc. 86 (2021), no. 5, 445–457.
[21] T.A. Selenge and B. Horoldagva, Extremal Kragujevac trees with respect to Sombor indices, Commun. Comb. Optim. 9 (2024), no. 1, 177–183.
[22] Y. Shang, Sombor index and degree-related properties of simplicial networks, Appl. Math. Comput. 419 (2022), Article ID: 126881.
[23] V.K. Singh, V.P. Tewari, D.K. Gupta, and A.K. Srivastava, Calculation of heat of formation:-Molecular connectivity and IOC-ω technique, a comparative study, Tetrahedron 40 (1984), no. 15, 2859–2863.–4020(01)91294–3