Analytical Study of Second Inverse Sum Indeg Index of Special Graphs

Articles in Press

Document Type : Original paper

Authors

1 Faculty of science, Mahallat Institute of Higher Education, Mahallat, Iran

2 Faculty of Science, University of Kragujevac, Kragujevac, Serbia

3 Department of Applied Science( Mathematics), Rajiv Gandhi Institute of Technology (RIT), Kottayam – 686501, India

Abstract

In this paper, we investigate the second inverse sum indeg index $ISI_2$ of graphs, a topological index that has significant applications in chemical graph theory. Upper and lower bounds for $ISI_2$ of graphs and trees with a specified number of pendent edges are established. Furthermore, $ISI_2$ of various bridge graphs are computed. The main contribution of this work lies in presenting precise bounds and exact expressions for particular families of graphs, offering resources for researchers and engineers in mathematical chemistry and applied graph theory.

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References

[1] L. Alex and I. Gopalapillai, On a conjecture on edge Mostar index of bicyclic graphs, Iranian J. Math. Chem. 14 (2023), no. 2, 97–108. https://doi.org/10.22052/ijmc.2023.248632.1680
[2] L. Alex and G. Indulal, On some extremal results and bounds of additively weighted edge Mostar index, Iranian J. Math. Chem. 16 (2025), no. 1, 51–64. https://doi.org/10.22052/ijmc.2024.254661.1858
[3] L. Alex and G. Indulal, Sharp bounds on additively weighted Mostar index of cacti, Commun. Comb. Optim. 11 (2026), no. 1, 117–130. https://doi.org/10.22049/cco.2024.28757.1702
[4] L. Alex, N. Pullanhiyodan, and H. K C, On the weighted bond additive indices of some nanostructures, J. Discrete Math. Appl. 9 (2024), no. 4, 289–307. https://doi.org/10.22061/jdma.2024.11216.1092
[5] A. Ali, T. Došlić, and Z. Raza, On trees of a fixed maximum degree with extremal general atom-bond sum-connectivity index, J. Appl. Math. Comput. 71 (2025), no. 1, 1035–1049. https://doi.org/10.1007/s12190-024-02275-1
[6] A. Ali, L. Zhong, and I. Gutman, Harmonic index and its generalizations: extremal results and bounds, MATCH Commun. Math. Comput. Chem. 81 (2019), no. 2, 249–311.
[7] R. Alidehi Ravandi and H. Omer Abdullah, Topological indices of drug molecular structures: An application with the treatment and prevention of Covid-19, J. Discrete Math. Appl. 8 (2023), no. 2, 81–101. https://doi.org/10.22061/jdma.2023.1944
[8] K.C. Das, On comparing Zagreb indices of graphs, MATCH Commun. Math. Comput. Chem. 63 (2010), no. 2, 433–440.
[9] E. Deutsch and S. Klavžar, M-polynomial and degree-based topological indices, Iranian J. Math. Chem. 6 (2015), no. 2, 93–102. https://doi.org/10.22052/ijmc.2015.10106
[10] G. Fath-Tabar, B. Furtula, and I. Gutman, A new geometric-arithmetic index, J. Math. Chem. 47 (2010), no. 1, 477–486.
https://doi.org/10.1007/s10910-009-9584-7
[11] M. Ghorbani and Z. Vaziri, On the Szeged and Wiener complexities in graphs, Appl. Math. Comput. 470 (2024), #128532. https://doi.org/10.1016/j.amc.2024.128532
[12] M. Ghorbani, Z. Vaziri, and M. Dehmer, Complexity measures in trees: a comparative investigation of Szeged and Wiener indices, Comput. Appl. Math. 44 (2025), no. 4, #195. https://doi.org/10.1007/s40314-025-03159-1
[13] A. Graovac and M. Ghorbani, A new version of atom-bond connectivity index, Acta Chim. Slov. 57 (2010), no. 3, 609–612.
[14] I. Gutman, Degree-based topological indices, Croatica Chemica Acta 86 (2013), no. 4, 351–361. http://doi.org/10.5562/cca2294
[15] I. Gutman and A.A. Dobrynin, The Szeged index–a success story, Graph Theory Notes N.Y. 34 (1998), 37–44.
[16] I. Gutman and O.E. Polansky, Mathematical concepts in organic chemistry, Springer Science and Business Media, 1986.
[17] D. Heidari, M.R. Rostami, I. Gutman, and L. Alex, New results on second inverse sum indeg index, Iranian J. Math. Chem. (2026), to appear.
[18] P.E. John, P.V. Khadikar, and J. Singh, A method of computing the PI index of benzenoid hydrocarbons using orthogonal cuts, J. Math. Chem. 42 (2007), no. 1, 37–45. https://doi.org/10.1007/s10910-006-9100-2
[19] P.V. Khadikar, On a novel structural descriptor PI, National Academy of Science Letters 23 (2000), 113–118.
[20] P.V. Khadikar, N.V. Deshpande, P.P. Kale, A. Dobrynin, I. Gutman, and G. Domotor, The Szeged index and an analogy with the Wiener index, Journal of Chemical Information and Computer Sciences 35 (1995), no. 3, 547–550. https://doi.org/10.1021/ci00025a024
[21] P.V. Khadikar, P.P. Kale, N.V. Deshpande, S. Karmarkar, and V.K. Agrawal, Novel PI indices of hexagonal chains, J. Math. Chem. 29 (2001), no. 2, 143–150. https://doi.org/10.1023/A:1010931213729
[22] P.V. Khadikar, S. Karmarkar, and V.K. Agrawal, A novel PI index and its applications to QSPR/QSAR studies, J. Chem. Inf. Comput. Sci. 41 (2001), no. 4, 934–949. https://doi.org/10.1021/ci0003092
[23] A.J.M. Khalaf, M.F. Hanif, M.K. Siddiqui, and M.R. Farahani, On degree based topological indices of bridge graphs, J. Discrete Math. Sci. Cryptogr. 23 (2020), no. 6, 1139–1156. https://doi.org/10.1080/09720529.2020.1822040
[24] H. Lin, A survey of recent extremal results on the Wiener index of trees, MATCH Commun. Math. Comput. Chem. 92 (2024), 253–270. https://doi.org/10.46793/match.92-2.253L
[25] O.M. Minailiuc, G. Katona, M.V. Diudea, M. Strunje, A. Graovac, and I. Gutman, Szeged fragmental indices, Croatica Chemica Acta 71 (1998), no. 3, 473–488.
[26] M. Rostami, M. Shabanian, and H. Moghanian, Some topological indices for theoretical study of two types of nanostar denderimers, Digest J. Nanomater. Biostruct 7 (2012), no. 1, 247–252. http://irdoi.ir/904-306-820-179
[27] M. Rostami and M. Sohrabi-Haghighat, Further results on new version of atombond connectivity index, MATCH Commun. Math. Comput. Chem. 71 (2014), no. 1, 21–32.
[28] M. Rostami, M. Sohrabi-Haghighat, and M. Ghorbani, On second atom-bond connectivity index., Iranian J. Math. Chem. 4 (2013), no. 2, 256–270. https://doi.org/10.22052/ijmc.2013.5302
[29] M. Sohrabi-Haghighat and M. Rostami, Using linear programming to find the extremal graphs with minimum degree 1 with respect to geometric-arithmetic index, Applied mathematics in Engineering, Management and Technology 3 (2015), no. 1, 534–539.
[30] M. Sohrabi-Haghighat and M. Rostami, Relations between second geometric-arithmetic index and second atom-
bond connectivity index, Int. J. Appl. Math. Stat. 57 (2018), no. 1, 73–81. http://irdoi.ir/540-738-069-130
[31] Z. Tang, Some properties of eccentrical graphs, Contrib. Math. 9 (2024), 18–25. https://doi.org/10.47443/cm.2024.006
[32] S. Wagner and H. Wang, Introduction to Chemical Graph Theory, Chapman and Hall/CRC, 2018.
[33] Y. Wu, C. Hong, P. Fu, and W. Lin, Large trees with maximal inverse sum indeg index have no vertices of degree 2 or 3, Discrete Appl. Math. 360 (2025), 131–138.  https://doi.org/10.1016/j.dam.2024.09.006
[34] Y. Yuan, B. Zhou, and N. Trinajstić, On geometric-arithmetic index, J. Math. Chem. 47 (2010), no. 2, 833–841.
https://doi.org/10.1007/s10910-009-9603-8
[35] I. Zhou, B. Gutman, B. Furtula, and Z. Du, On two types of geometric–arithmetic index, Chemical Physics Lett. 482 (2009), no. 1-3, 153–155. https://doi.org/10.1016/j.cplett.2009.09.102
Communications in Combinatorics and Optimization

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Available Online from 29 January 2026

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