AIMS Mathematics, 2020, 5(5): 4085-4107. doi: 10.3934/math.2020262.

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On irregularity descriptors of derived graphs

1 School of Information Science and Technology, Yunnan Normal University, Kunming 650500, China
2 School of Natural Sciences, National University of Sciences and Technology, Islamabad 44000, Pakistan
3 Department of Mathematics and Statistics, Institute of Southern Punjab, Multan 66000, Pakistan
4 University of Engineering and Technology, Lahore (RCET) 54000, Pakistan

Topological indices are molecular structural descriptors which computationally and theoretically describe the natures of the underlying connectivity of nanomaterials and chemical compounds, and hence they provide quicker methods to examine their activities and properties. Irregularity indices are mainly used to characterize the topological structures of irregular graphs. Graph irregularity studies are useful not only for quantitative structure-activity relationship (QSAR) and quantitative structure-property relationship (QSPR) studies, but also for predicting their various physical and chemical properties, including toxicity, resistance, melting and boiling points, the enthalpy of evaporation and entropy. In this article, we establish the expressions for the irregularity indices named as the variance of vertex degrees, σ irregularity index, and the discrepancy index of subdivision graph, vertex-semi total graph, edge-semi total graph, total graph, line graph, paraline graph, double graph, strong double graph and extended double cover of a graph.
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Keywords irregularity indices; degree; total graph; double graph

Citation: Wei Gao, Zahid Iqbal, Shehnaz Akhter, Muhammad Ishaq, Adnan Aslam. On irregularity descriptors of derived graphs. AIMS Mathematics, 2020, 5(5): 4085-4107. doi: 10.3934/math.2020262


  • 1. P. Liu, W. Long, Current mathematical methods used in QSAR/QSPR studies, Int. J. Mol. Sci., 10 (2009), 1978-1998.    
  • 2. T. Réti, R. Sharafdini, A. Dregelyi-Kiss, et al. Graph irregularity indices used as molecular descriptor in QSPR studies, Match Commun. Math. Comput. Chem., 79 (2018), 509-524.
  • 3. S. Akhter, M. Imran, Computing the forgotten topological index of four operations on graphs, AKCE Int. J. Graphs Comb., 14 (2017), 70-79.    
  • 4. S. Akhter, M. Imran, The sharp bounds on general sum-connectivity index of four operations on graphs, J. Inequal. Appl., 2016 (2016), 241.
  • 5. S. Akhter, M. Imran, W. Gao, et al. On topological indices of honeycomb networks and graphene networks, Hac. J. Math. Stat., 47 (2018), 19-35.
  • 6. S. Akhter, Two degree distance based topological indices of chemical trees, IEEE Access, 7 (2019), 95653-95658.    
  • 7. H. Abdo, D. Dimitrov, W. Gao, On the irregularity of some molecular structures, Can. J. Chem., 95 (2017), 174-183.    
  • 8. I. Gutman, P. Hansen, H. Mélot, Variable neighborhood search for extremal graphs 10. Comparison of irregularity indices for chemical trees, J. Chem. Inf. Model., 45 (2005), 222-230.    
  • 9. H. Yang, M. Imran, S. Akhter, et al. On distance-based topological descriptors of subdivision vertex-edge join of three graphs, IEEE Access, 7 (2019), 143381-143391.    
  • 10. J. Zheng, Z. Iqbal, A. Fahad, et al. Some eccentricity-based topological indices and polynomials of poly(EThyleneAmidoAmine) (PETAA) dendrimers, Processes, 7 (2019), 433.
  • 11. Z. Iqbal, M. Ishaq, A. Aslam, et al. On eccentricity-based topological descriptors of water-soluble dendrimers, Z. Naturforsch. C, 74 (2018), 25-33.
  • 12. W. Gao, Z. Iqbal, M. Ishaq, et al. Bounds on topological descriptors of the corona product of F-sum of connected graphs, IEEE Access, 7 (2019), 26788-26796.    
  • 13. W. Gao, Z. Iqbal, M. Ishaq, et al. Topological aspects of dendrimers via distancebased descriptors, IEEE Access, 7 (2019), 35619-35630.    
  • 14. Z. Iqbal, M. Ishaq, M. Aamir, On eccentricity-based topological descriptors of dendrimers, Iran. J. Sci. Technol. A, 43 (2019), 1523-1533.    
  • 15. S. Hayat, M. Imran, J. B. Liu, Correlation between the Estrada index and p-electronic energies for benzenoid hydrocarbons with applications to boron nanotubes, Int. J. Quantum Chem., 119 (2019), e26016.
  • 16. S. Hayat, M. Imran, J. B. Liu, An efficient computational technique for degree and distance based topological descriptors with applications, IEEE Access, 7 (2019), 32276-32296.    
  • 17. S. Hayat, S. Wang, J. B. Liu, Valency-based topological descriptors of chemical networks and their applications, Appl. Math. Model., 60 (2018), 164-178.    
  • 18. S. Hayat, Computing distance-based topological descriptors of complex chemical networks: New theoretical techniques, Chem. Phys. Lett., 688 (2017), 51-58.    
  • 19. M. O. Albertson, The irregularity of a graph, Ars Combinatoria, 46 (1996), 219-225.
  • 20. I. Gutman, N. Trinajstić, Graph theory and molecular orbitals, Total π electron energy of alternant hydrocarbons, Chem. Phys. Lett., 17 (1972), 535-538.    
  • 21. A. Furtula, I. Gutman, A forgotten topological index, J. Math. Chem., 53 (2015), 1184-1190.    
  • 22. I. Gutman, M. Togan, A. Yurttas, et al. Inverse problem for σ index, Match Commun. Math. Comput. Chem., 79 (2018), 491-508.
  • 23. H. Abdo, D. Dimitrov, I. Gutman, Graphs with maximal σ irregularity, Discrete Appl. Math., 250 (2018), 57-64.    
  • 24. I. Gutman, Stepwise irregular graphs, Appl. Math. Comput., 325 (2018), 234-238.
  • 25. F. K. Bell, A note on the irregularity of graphs, Linear Algebra Appl., 161 (1992), 45-54.    
  • 26. J. Haviland, On irregularity in graphs, Ars Combinatoria, 78 (2006), 283-288.
  • 27. C. J. Lawrence, K. Tizzard, J. Haviland, Disease-spread and stochastic graphs, Proc. International Conference on Social Networks, London, (1995), 143-150.
  • 28. W. Gao, M. Aamir, Z. Iqbal, et al. On irregularity measures of some dendrimers structures, Mathematics, 7 (2019), 271.
  • 29. Z. Iqbal, A. Aslam, M. Ishaq, et al. Characteristic study of irregularity measures of some nanotubes, Can. J. Phys., 97 (2019), 1125-1132.    
  • 30. D. Zhao, Z. Iqbal, R. Irfan, et al. Comparison of irregularity indices of several dendrimers structures, Processes, 7 (2019), 662.
  • 31. N. De, A. Pal, S. M. A. Nayeem, The irregularity of some composite graphs, Int. J. Appl. Comput. Math., 2 (2016), 411-420.    
  • 32. M. Tavakoli, F. Rahbarnia, A. R. Ashrafi, Some new results on irregularity of graphs, J. Appl. Math. Inform., 32 (2014), 675-685.    
  • 33. H. Abdo, D. Dimitrov, The total irregularity of some composite graphs, Int. J. Comput. Appl., 122 (2015), 0975-8887.
  • 34. S. M. Hosamani, I. Gutman, Zagreb indices of transformation graphs and total transformation graphs, Appl. Math. Comput., 247 (2014), 1156-1160.


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