Intuitionistic fuzzy Sombor indices: A novel approach for improving the performance of vaccination centers

Document Type : Original paper


1 Department of Mathematics, Riphah International University, Lahore, Pakistan

2 University of Kragujevac


Intuitionistic fuzzy graphs are generalizations of fuzzy graphs, in which each vertex is assigned an ordered pair whose first coordinate gives the membership value and the second coordinate gives the non-membership value. There are many theoretical parameters to study different types of graphs and fuzzy graphs, topological indices are one of them. Sombor indices are important in explaining the topology of a graph, and were found to possess useful applicative properties. The two versions of the Sombor indices ($SO_3$ and $SO_4$)are converted into an intuitionistic fuzzy framework, and then formulas for different kinds of graphs are calculated. Our study also involves setting up a network of vaccination centers during a pandemic and applying intuitionistic fuzzy-based topological indices to assess their performance. With the help of this application, we highlight the practical implication and benefits of employing intuitionistic fuzzy-based techniques in vaccination centers. Through a comparative analysis, we evaluate which index is more efficient.


Main Subjects

[1] R. Aguilar-Sánchez, J.A. Méndez-Bermúdez, J.M. RodrÍguez, and J.M. Sigarreta, Normalized Sombor indices as complexity measures of random networks, Entropy 23 (2021), no. 8, Article ID: 976.
[2] U. Ahmad, S. Ahmad, and R. Yousaf, Computation of Zagreb and atom-bond connectivity indices of certain families of dendrimers by using automorphism group action, J. Serb. Chem. Soc. 82 (2017), no. 2, 151–162.
[3] U. Ahmad and T. Batool, Domination in rough fuzzy digraphs with application, Soft Computing 27 (2023), no. 5, 2425–2442.
[4] U. Ahmad, N.K. Khan, and A.B. Saeid, Fuzzy topological indices with application to cybercrime problem, Granul. Comput. 8 (2023), no. 5, 967–980.
[5] U. Ahmad and I. Nawaz, Directed rough fuzzy graph with application to trade networking, Compute. Appl. Math. 41 (2022), no. 8, Article ID: 366.
[6] U. Ahmad and I. Nawaz, Wiener index of a directed rough fuzzy graph and application to human trafficking, J. Intell. Fuzzy Syst. 44 (2023), no. 1, 1479–1495.
[7] U. Ahmad, I. Nawaz, and S. Broumi, Connectivity index of directed rough fuzzy graphs and its application in traffic flow network, Granul. Comput. 8 (2023), no. 6, 1561–1582.
[8] M. Akram, Bipolar fuzzy graphs, Inf. Sci. 181 (2011), no. 24, 5548–5564.
[9] M. Akram, U. Ahmad, and Rukhsar, Threshold graphs under picture Dombi fuzzy information, Granul. Comput. 7 (2022), no. 3, 691–707.
[10] M. Akram and N.O. Alshehri, Intuitionistic fuzzy cycles and intuitionistic fuzzy trees, Sci. World J. 2014 (2014), Article ID: 305836.
[11] M. Akram and M. Arshad, A new approach based on fuzzy rough digraphs for decision-making, J. Intell. Fuzzy Syst. 35 (2018), no. 2, 2105–2121.
[12] M. Akram and B. Davvaz, Strong intuitionistic fuzzy graphs, Filomat 26 (2012), no. 1, 177–196.
[13] M. Akram and W.A. Dudek, Intuitionistic fuzzy hypergraphs with applications, Inf. Sci. 218 (2013), 182–193.
[14] M. Akram, K. Ullah, G. Ćirović, and D. Pamucar, Algorithm for energy resource selection using priority degree-based aggregation operators with generalized orthopair fuzzy information and aczel–alsina aggregation operators, Energies 16
(2023), no. 6, Article ID: 2816.
[15] M. Akram and F. Zafar, A new approach to compute measures of connectivity in rough fuzzy network models, J. Intell. Fuzzy Syst. 36 (2019), no. 1, 449–465.
[16] M. Akram and F. Zafar, Rough fuzzy digraphs with application, J. Appl. Math. Comput. 59 (2019), no. 1, 91–127.
[17] T. Al-Hawary, Complete fuzzy graphs, Int. J. Math. Comput. Sci. 4 (2011), 26–34.
[18] F. Asif, Z. Zahid, M.N. Husin, M. Cancan, Z. Ta¸s, M. Alaeiyan, and M.R. Farahani, On Sombor indices of line graph of silicate carbide $Si2C3−I[p, q]$, J. Discrete Math. Sci. Cryptogr. 25 (2022), no. 1, 301–310.
[19] K.T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets Syst. 20 (1986), no. 1, 87–96.
[20] M. Azeem, M. Imran, and M.F. Nadeem, Sharp bounds on partition dimension of hexagonal Möbius ladder, J. King Saud Univ. Sci. 34 (2022), no. 2, Article ID:101779.
[21] M. Azeem, M.K. Jamil, and Y. Shang, Notes on the localization of generalized hexagonal cellular networks, Mathematics 11 (2023), no. 4, Article ID: 844.
[22] M. Azeem and M.F. Nadeem, Metric-based resolvability of polycyclic aromatic hydrocarbons, Eur. Phys. J. Plus 136 (2021), no. 4, Article ID: 395.
[23] P. Bhattacharya, Some remarks on fuzzy graphs, Pattern Recognit. Lett. 6 (1987), no. 5, 297–302.
[24] K.R. Bhutani, On automorphisms of fuzzy graphs, Pattern Recognit. Lett. 9 (1989), no. 3, 159–162.
[25] K.R Bhutani and A. Rosenfeld, Geodesies in fuzzy graphs, Electron. Notes Discrete Math. 15 (2003), 49–52.
[26] M. Binu, S. Mathew, and J.N. Mordeson, Connectivity index of a fuzzy graph and its application to human trafficking, Fuzzy Sets Syst. 360 (2019), 117–136.
[27] M. Binu, S. Mathew, and J.N. Mordeson, Wiener index of a fuzzy graph and application to illegal immigration networks, Fuzzy Sets Syst. 384 (2020), 132–147.
[28] K.C. Das and Y. Shang, Some extremal graphs with respect to Sombor index, Mathematics 9 (2021), no. 11, Article ID: 1202.
[29] J. Dinar, Z. Hussain, S. Zaman, and S.U. Rehman, Wiener index for an intuitionistic fuzzy graph and its application in water pipeline network, Ain Shams Eng. J. 14 (2023), no. 1, Article ID:101826.
[30] A.E.F. A. El-Atik, M.K. El-Bably, and M.A. El-Gayar, Topological visualization of rough sets by neighborhoods, a heart application based graphs,  (2022) Manuscript.
[31] A.O. Esogbue, M. Theologidu, and K. Guo, On the application of fuzzy sets theory to the optimal flood control problem arising in water resources systems, Fuzzy Sets Syst. 48 (1992), no. 2, 155–172.
[32] A.N. Gani and S.S. Begum, Degree, order and size in intuitionistic fuzzy graphs, Int. J. Algorithms Comput. Math. 3 (2010), no. 3, 11–16.
[33] A.N. Gani and S.R. Latha, On irregular fuzzy graphs, Appl. Math. Sci. 6 (2012), no. 11, 517–523.
[34] W. Gao, J.L.G. Guirao, and H. Wu, Nordhaus–Gaddum type inequalities for some distance-based indices of bipartite molecular graphs, J. Math. Chem. 58 (2020), no. 7, 1345–1352.
[35] A. Garrido, A brief history of fuzzy logic, Broad Research in Artificial Intelligence and Neuroscience 3 (2012), no. 1, 71–77.
[36] S. Gong and G. Hua, Remarks on wiener index of bipolar fuzzy incidence graphs, Front. Phys. 9 (2021), Article ID: 677882.
[37] S. Gong and G. Hua, Topological indices of bipolar fuzzy incidence graph, Open Chem. 19 (2021), no. 1, 894–903.
[38] I. Gutman, Geometric approach to degree-based topological indices: Sombor indices, MATCH Commun. Math. Comput. Chem. 86 (2021), no. 1, 11–16.
[39] I. Gutman, Some basic properties of Sombor indices, Open J. Discret. Appl. Math. 4 (2021), no. 1, 1–3.
[40] I. Gutman, N.K. Gürsoy, A. Gürsoy, and A. Ülker, New bounds on Sombor index, Commun. Comb. Optimi. 8 (2023), no. 2, 305–311.
[41] A. Hussain, K. Ullah, M. Mubasher, T. Senapati, and S. Moslem, Interval-valued
pythagorean fuzzy information aggregation based on Aczel-Alsina operations and
their application in multiple attribute decision making, IEEE Access 11 (2023),
[42] A. Hussain, H. Wang, K. Ullah, H. Garg, and D. Pamucar, Maclaurin symmetric mean aggregation operators based on novel frank T-norm and T-conorm for intuitionistic fuzzy multiple attribute group decision-making, Alex. Eng. J. 71 (2023),
[43] M. Imran, R. Luo, M.K. Jamil, M. Azeem, and K.M. Fahd, Geometric perspective to degree–based topological indices of supramolecular chain, Results in Engineering 16 (2022), Article ID: 100716.
[44] S.R. Islam and M. Pal, Hyper-Wiener index for fuzzy graph and its application in share market, J. Intell. Fuzzy Syst. 41 (2021), no. 1, 2073–2083.
[45] S.R. Islam and M. Pal, Further development of f-index for fuzzy graph and its application in indian railway crime, J. Appl. Math. Comput. 69 (2023), no. 1, 321–353.
[46] K. Jabeen, K. Ullah, M. Akram, and I. Haleemzai, Interval valued picture fuzzy Aczel–Alsina aggregation operators and their application by using the multiattribute decision making problem, J. Math. 2023 (2023), Article ID:1707867.
[47] S. Kalathian, S. Ramalingam, S. Raman, and N. Srinivasan, Some topological indices in fuzzy graphs, J. Intell. Fuzzy Syst. 39 (2020), no. 5, 6033–6046.
[48] M.G. Karunambigai, M. Akram, S. Sivasankar, and K. Palanivel, Balanced intuitionistic fuzzy graphs, Appl. Math. Sci. 7 (2013), no. 51, 2501–2514.
[49] S. Kosari, N. Dehgardi, and A. Khan, Lower bound on the KG-sombor index, Commun. Comb. Optim. 8 (2023), no. 4, 751–757.
[50] H. Liu, I. Gutman, L. You, and Y. Huang, Sombor index: review of extremal results and bounds, J. Math. Chem. 60 (2022), 771–798.
[51] I. Masmali, A. Ahmad, M. Azeem, and A.N.A. Koam, Madm and assessment of pilot health projects based on spherical fuzzy information, Neural. Comput. Appl. 35 (2023), 16619–16632.
[52] S. Mathew and M.S. Sunitha, Types of arcs in a fuzzy graph, Inf. Sci. 179 (2009), no. 11, 1760–1768
[53] J.N. Mordeson, Fuzzy line graphs, Pattern Recognit. Lett. 14 (1993), no. 5, 381–384.
[54] J.N. Mordeson and P. Chang-Shyh, Operations on fuzzy graphs, Inf. Sci. 79 (1994), no. 3-4, 159–170.
[55] F. Movahedi and M.H. Akhbari, Degree-based topological indices of the molecular structure of hyaluronic acid–methotrexate conjugates in cancer treatment, Int. J. Quantum Chem. 123 (2023), no. 12, Article ID: e27106.
[56] R. Parvathi and M. G. Karunambigai, Intuitionistic Fuzzy Graphs, Computational Intelligence, Theory and Applications (B. Reusch, ed.), vol. 38, Springer Berlin Heidelberg, Berlin, Heidelberg, 2006, pp. 139–150.
[57] S. Poulik and G. Ghorai, Determination of journeys order based on graph’s wiener absolute index with bipolar fuzzy information, Inf. Sci. 545 (2021), 608–619.
[58] I. Redžepović, Chemical applicability of sombor indices, J. Serb. Chem. Soc. 86 (2021), no. 5, 445–457.
[59] T.A. Selenge and B. Horoldagva, Extremal Kragujevac trees with respect to Sombor indices, Commun. Comb. Optim. 9 (2024), no. 1, 177–183.
[60] Y. Shang, Sombor index and degree-related properties of simplicial networks, Appl. Math. Comput. 419 (2022), Article ID: 126881.
[61] A.A. Shashidharaa, H. Ahmed, S. Nandappa, and M. Cancanc, Domination version: Sombor index of graphs and its significance in predicting physicochemical properties of butane derivatives, Eurasian Chem. Commun. 5 (2023), no. 1, 91–
[62] M.S. Sunitha and S. Mathew, Fuzzy graph theory: a survey, Ann. Pure Appl. Math. 4 (2013), no. 1, 92–110.
[63] H. Wang, T. Xu, L. Feng, T. Mahmood, and K. Ullah, Aczel–Alsina Hamy mean aggregation operators in T-spherical fuzzy multi-criteria decision-making, Axioms 12 (2023), no. 2, Article ID: 224.
[64] B.K. Wong and V.S. Lai, A survey of the application of fuzzy set theory in production and operations management: 1998–2009, Int. J. Prod. Econ. 129 (2011), no. 1, 157–168.
[65] L.A. Zadeh, Fuzzy sets, Inf. Control. 8 (1965), no. 3, 338–353.
[66] L.A. Zadeh, Making computers think like people [fuzzy set theory], IEEE spectrum 21 (1984), no. 8, 26–32.
[67] H.-J. Zimmermann, Fuzzy set theory, Wiley Interdiscip Rev. Comput. Stat. 2 (2010), no. 3, 317–332.