Second Zagreb index for fuzzy graphs and its application in mathematical chemistry

Document Type : Research Paper

Authors

Department of Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University, Midnapore 721102, India

Abstract

The Zagreb index (ZI) of a crisp graph and also for a fuzzy graph (FG) is a very much useful tool in network theory, spectral graph theory, molecular chemistry and several fields of chemistry and mathematics. The second ZI is studied for FGs here. Bounds of this index are calculated for several FGs: path, star, cycle, complete FG, partial fuzzy subgraph, etc. For isomorphic FGs, it is shown that the value of this index is same. Bounds of this index for the Cartesian product, composition, join and union of two FGs are established. At the end of this article, an application of the index in mathematical chemistry is studied. For this, octane isomers are considered and analyzed the correlation between this index with some physico-chemical properties of octane isomers. This index's correlation coefficient ($r$) with acentric factor and entropy is determined for the linear curve fittings. Using the value of $r$, one can conclude that this index can help estimate the acentric factor and entropy with significant accuracy. Also, these outcomes declare the  appropriateness of the index in QSPR research.

Keywords

References

[1] M. Akram, Bipolar fuzzy graphs, Information Sciences, 181(24) (2011), 5548-5564.
[2] M. Akram, m-polar fuzzy graphs, theorem, methods, application, Springer, Berlin, (2019). DOI:10.1007/978-3-030-03751-2.
[3] M. Akram, U. Ahmad, Rukhsar, F. Karaaslan, Complex Pythagorean fuzzy threshold graphs with application in petroleum replenishment, Journal of Applied Mathematics and Computing, 68 (2022), 2125-2150.
[4] M. Akram, S. Alsulami, F. Karaaslan, A. Khan, q-Rung orthopair fuzzy graphs under Hamacher operators, Journal of Intelligent and Fuzzy Systems, 40(1) (2021), 1367-1390.
[5] M. Akram, S. Nawaz, On fuzzy soft graphs, Italian Journal of Pure and Applied Mathematics, 34 (2015), 497-514.
[6] M. Akram, F. Zafar, A new approach to compute measures of connectivity in rough fuzzy network models, Journal of Intelligent and Fuzzy Systems, 36(1) (2019), 449-465.
[7] M. Binu, S. Mathew, J. N. Mordeson, Connectivity index of a fuzzy graph and its application to human trafficking, Fuzzy Sets and Systems, 360 (2019), 117-136.
[8] M. Binu, S. Mathew, J. N. Mordeson, Wiener index of a fuzzy graph and application to illegal immigration networks, Fuzzy Sets and Systems, 384 (2020), 132-147.
[9] M. Binu, S. Mathew, J. N. Mordeson, Cyclic connectivity index of fuzzy graphs, IEEE Transactions on Fuzzy Systems, 29(6) (2021), 1340-1349.
[10] M. Binu, S. Mathew, J. N. Mordeson, Connectivity status of fuzzy graphs, Information Sciences, 573 (2021), 382-395.
[11] R. A. Borzooei, H. Rashmanlou, Domination in vague graphs and its applications, Journal of Intelligent and Fuzzy Systems, 29(5) (2015), 1933-1940.
[12] K. C. Das, I. Gutman, Some properties of the second Zagreb index, MATCH Communications in Mathematical and in Computer Chemistry, 52 (2004), 103-112.
[13] B. Fortula, I. Gutman, A forgotten topological index, Journal of Mathematical Chemistry, 53(4) (2015), 1184-1190.
[14] I. Gutman, N. Trinajstic, Graph theorem and molecular orbitals. Total π-electron energy of alternant hydrocarbons, Chemical Physics Letters, 17 (1972), 535-538.
[15] S. Hameed, M. Akram, N. Mustafa, F. Karaaslan, Extension of threshold graphs under complex intuitionistic fuzzy environment, Journal of Multiple-Valued Logic and Soft Computing, 37(3-4) (2021), 295-315.
[16] K. Hayat, B. Y. Cao, M. I. Ali, F. Karaaslan, Z. Qin, Characterizations of certain types of type 2 soft graphs, Discrete Dynamics in Nature and Society, (2018). DOI:10.1155/2018/8535703.
[17] S. R. Islam, S. Maity, M. Pal, Comment on “ Wiener index of a fuzzy graph and application to illegal immigration networks”, Fuzzy Sets and Systems, 384 (2020), 148-151.
[18] S. R. Islam, M. Pal, First Zagreb index on a fuzzy graph and its application, Journal of Intelligent and Fuzzy Systems, 40(6) (2021), 10575-10587.
[19] S. R. Islam, M. Pal, Hyper-Wiener index for fuzzy graph and its application in share market, Journal of Intelligent and Fuzzy Systems, 41(1) (2021), 2073-2083.
[20] S. R. Islam, M. Pal, Hyper-connectivity index for fuzzy graph with application, TWMS, Journal of Applied and Engineering Mathematics, (2021), to appear.
[21] S. R. Islam, M. Pal, F-index for fuzzy graph with application, TWMS Journal of Applied and Engineering Mathematics, (2021), to appear.
[22] S. R. Islam, M. Pal, Further development of F-index for fuzzy graph and its application in Indian railway crime, Journal of Applied Mathematics and Computing, (2022). DOI:10.1007/s12190-022-01748-5.
[23] M. Javaid, M. Ibraheem, U. Ahmad, J. B. Liu, Sharp bounds of first Zagreb coindex for F-sum graphs, Journal of Mathematics, (2021). DOI:10.1155/2021/9984412.
[24] A. Josy, S. Mathew, J. N. Mordeson, Neighborhood connectivity index of a fuzzy graph and its application to human trafficking, Iranian Journal of Fuzzy Systems, 19(3) (2022), 139-154.
[25] S. Kalathian, S. Ramalingam, S. Raman, N. Srinivasan, Some topological indices in fuzzy graphs, Journal of Intelligent and Fuzzy Systems, 39(5) (2020), 6033-6046.
[26] F. Karaaslan, Hesitant fuzzy graphs and their applications in decision making, Journal of Intelligent and Fuzzy Systems, 36(3) (2019), 2729-2741.
[27] F. Karaaslan, B. Davvaz, Properties of single-valued neutrosophic graphs, Journal of Intelligent and Fuzzy Systems, 34(1) (2018), 57-79.
[28] R. Kazemi, The second Zagreb index of molecular graphs with tree structure, MATCH Communications in Mathematical and in Computer Chemistry, 72(3) (2014), 753-760.
[29] M. H. Khalifeh, H. Y. Azari, A. R. Ashrafi, The first and second Zagreb indices of some graph operations, Discrete Applied Mathematics, 157(4) (2009), 804-811.
[30] E. Konstantinova, The discrimination ability of some topological and information distance indices for graphs of unbranched hexagonal systems, Journal of Chemical Information and Computer Sciences, 36(1) (1996), 54-57.
[31] R. Mahapatra, S. Samanta, M. Pal, Generalized neutrosophic planer graphs and its application, Journal of Applied Mathematics and Computing, (2020). DOI:10.1007/s12190-020-01411-x.
[32] R. Mahapatra, S. Samanta, M. Pal, Q. Xin, RSM index: A new way of link prediction in social networks, Journal of Intelligent and Fuzzy Systems, 37(2) (2019), 2137-2151.
[33] R. Mahapatra, S. Samanta, M. Pal, Q. Xin, Link prediction in social networks by neutrosophic graph, International Journal of Computational Intelligence Systems, 13(1) (2020), 1699-1713.
[34] S. Mathew, J. N. Mordeson, H. L. Yang, Incidence cuts and connectivity in fuzzy incidence graphs, Iranian Journal of Fuzzy Systems, 16(2) (2019), 31-43.
[35] S. Mondal, N. De, A. Pal, On neighborhood Zagreb index of product graphs, Journal of Molecular Structure, (2021).
[36] J. N. Mordeson, S. Mathew, Advanced topics in fuzzy graph theory, Springer, 2019.
[37] G. Muhiuddin, S. Hameed, A. Maryam, U. Ahmad, Cubic Pythagorean fuzzy graphs, Journal of Mathematics, (2022). DOI:10.1155/2022/1144666.
[38] G. Muhiuddin, S. Hameed, A. Rasheed, U. Ahmad, Cubic planar graph and its application to road network, Mathematical Problems in Engineering, (2022). DOI:10.1155/2022/5251627.
[39] M. Pal, S. Samanta, G. Ghorai, Modern trends in fuzzy graph theory, Springer, Singapore, (2020). DOI:10.1007/978-981-15-8803-7.
[40] S. Poulik, S. Das, G. Ghorai, Randic index of bipolar fuzzy graphs and its application in network systems, Journal of Applied Mathematics and Computing, (2021). DOI:10.1007/s12190-021-01619-5.
[41] S. Poulik, G. Ghorai, Certain indices of graphs under bipolar fuzzy environment with applications, Soft Computing, 24 (2020), 5119-5131.
[42] S. Poulik, G. Ghorai, Determination of journeys order based on graph’s Wiener absolute index with bipolar fuzzy information, Information Sciences, 545 (2021), 608-619.
[43] H. Rashmanlou, R. A. Borzooei, Product vague graphs and its applications, Journal of Intelligent and Fuzzy Systems, 30(1) (2016), 371-382.
[44] H. Rashmanlou, S. Samanta, M. Pal, R. A. Borzooei, Product of bipolar fuzzy graphs and their degree, International Journal of General Systems, 45(1) (2016), 1-14.
[45] A. Rosenfield, Fuzzy sets and their applications, Academic Press, New York, (1975), 77-95.
[46] S. Samanta, M. Pal, Telecommunication system based on fuzzy graphs, Journal of Telecommunications System and Management, 3(1) (2013), 1-6.
[47] S. Samanta, M. Pal, A new approach to social networks based on fuzzy graphs, Turkish Journal of Fuzzy System, 5(2) (2014), 78-99.
[48] M. Sarwar, M. Akram, S. Shahzadi, Bipolar fuzzy soft information applied to hypergraphs, Soft Computing, 25 (2021), 3417-3439.
[49] S. Siddique, U. Ahmad, M. Akram, A decision-making analysis with generalized m-polar fuzzy graphs, Journal of Multiple-valued Logic and Soft Computing, 37 (2021), 409-436.
[50] H. Wiener, Structural determination of paraffin boiling points, Journal of the American Chemical Society, 69 (1947), 17-20.
[51] L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338-353.
Iranian Journal of Fuzzy Systems
Volume 20, Issue 1
January and February 2023
Pages 119-136

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