Bounds on Sombor index and inverse sum indeg (ISI) index of graph operations

Document Type : Original paper


1 Department of Mathematical Sciences, College of Science, United Arab Emirate University, Al Ain 15551, Abu Dhabi, UAE

2 Department of Mathematical Sciences, United Arab Emirates University, P.O. Box 15551, Al Ain, United Arab Emirates


Let $ G $ be a graph with vertex set $ V(G) $ and edge set $ E(G) $. Denote by $ d_G(u) $ the degree of a vertex $ u \in V(G) $. The Sombor index of $ G $ is defined as $ SO(G) = \sum_{uv \in E(G)} \sqrt{d_u^2 + d_v^2} $, whereas, the inverse sum indeg $ (ISI) $ index is defined as $ ISI(G) = \sum_{uv \in E(G)}    \frac{d_{u}d_{v}}{d_{u} + d_{v}}. $ In this paper, we compute the bounds in terms of maximum degree, minimum degree, order and size of the original graphs $ G $ and $ H $ for Sombor and $ ISI $ indices of several graph operations like corona product, cartesian product, strong product, composition and join of graphs.


Main Subjects

[1] S. Akhter and R. Farooq, Computing bounds for the general sum-connectivity index of some graph operations, Algebra Discrete Math. 29 (2020), no. 2, 147–160.
[2] S. Akhter, R. Farooq, and S. Pirzada, Exact formulae of general sum-connectivity index for some graph operations, Mat. Vesnik 70 (2018), no. 3, 267–282.
[3] A. Altassan, B.A. Rather, and M. Imran, Inverse sum indeg index (energy) with applications to anticancer drugs, Mathematics 10 (2022), no. 24, Article ID: 4749.
[4] R. Cruz, I. Gutman, and J. Rada, Sombor index of chemical graphs, Appl. Math. Comput. 399 (2021), Article ID: 126018.
[5] I. Gutman, Geometric approach to degree-based topological indices: Sombor indices, MATCH Commun. Math. Comput. Chem. 86 (2021), no. 1, 11–16.
[6] I. Gutman, Some basic properties of Sombor indices, Open J. Discrete Appl. Math. 4 (2021), no. 1, 1–3.–odam2021.0047
[7] I. Gutman and B. Furtula (Eds.), Novel Molecular Structure Descriptors–Theory and Applications I, Univ. Kragujevac, Kragujevac, 2010.
[8] I. Gutman, N.K. G¨ursoy, A. G¨ursoy, and A. ¨Ulker, New bounds on Sombor index, Commun. Comb. Optim. 8 (2023), no. 2, 305–311.
[9] I. Gutman and N. Trinajstić, Graph theory and molecular orbitals. total ϕ-electron energy of alternant hydrocarbons, Chem. Phys. Lett. 17 (1972), no. 4, 535–538.–2614(72)85099–1
[10] Ö. Havare, The inverse sum indeg index (ISI) and ISI energy of Hyaluronic Acid-Paclitaxel molecules used in anticancer drugs, Open J. Discrete Appl. Math. 4 (2021), no. 3, 72–81.–odam2021.0065
[11] S.M. Hosamani, B.B. Kulkarni, R.G. Boli, and V.M. Gadag, QSPR analysis of certain graph theocratical matrices and their corresponding energy, Appl. Math. Nonlinear Sci. 2 (2017), no. 1, 131–150.
[12] F. Jamal, M. Imran, and B.A. Rather, On inverse sum indeg energy of graphs, Special Matrices 11 (2023), no. 1, Article ID: 20220175.–2022–0175
[13] M.H. Khalifeh, H. Yousefi-Azari, and A.R. Ashrafi, The hyper-Wiener index of graph operations, Comput. Math. Appl. 56 (2008), no. 5, 1402–1407.
[14] M.H. Khalifeh, H. Yousefi-Azari, and A.R. Ashrafi, The first and second Zagreb indices of some graph operations, Discrete Appl. Math. 157 (2009), no. 4, 804–811.
[15] S.A.K. Kirmani, P. Ali, F. Azam, and P.A. Alvi, On ve-degree and ev-degree topological properties of hyaluronic acid-anticancer drug conjugates with QSPR, J. Chem. 2021 (2021), Artice ID: 3860856.
[16] V.R. Kulli, Sombor indices of certain graph operators, Int. J. Eng. Sci. Research Tec. 10 (2021), no. 1, 127–134.
[17] I. Milovanović, E. Milovanović, A. Ali, and M. Matejić, Some results on the Sombor indices of graphs, Contrib. Math. 3 (2021), 59–67.
[18] I. Redžepović, Chemical applicability of Sombor indices, J. Serb. Chem. Soc. 86 (2021), no. 5, 445–457.–0003–4956–0407
[19] T.A. Selenge and B. Horoldagva, Extremal Kragujevac trees with respect to Sombor indices, Commun. Comb. Optim., In press.
[20] B. Shwetha Shetty, V. Lokesha, and P.S. Ranjini, On the harmonic index of graph operations, Trans. Comb. 4 (2015), no. 4, 5–14.
[21] R. Todeschini and V. Consonni, Handbook of Molecular Descriptors, John Wiley & Sons, 2008.
[22] D. Vukicevic and M. Gasperov, Bond additive modeling 1. Adriatic indices, Croatica Chemica Acta 83 (2010), no. 3, 243–260.
[23] Z. Wang, Y. Mao, Y. Li, and B. Furtula, On relations between Sombor and other degree-based indices, J. Appl. Math. Comput. 68 (2022), 1–17.–021–01516–x
[24] S. Zangi, M. Ghorbani, and M. Eslampour, On the eigenvalues of some matrices based on vertex degree, Iranian J. Math. Chem. 9 (2018), no. 2, 149–156.