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On Topological Indices of Graph Transformation

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Abstract

In this paper, the exact formulae for the generalized degree distance, reciprocal degree distance and degree distance of Mycielskian graph and the complement of the Mycielskian graph are obtained. Also, we obtain the sharp bounds for the ABC index, GA index, third Zagreb index and third Zagreb coindex of the Mycielskian graph.

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References

  1. Alizadeh, Y., Iranmanesh, A., Doslic, T.: Additively weighted Harary index of some composite graphs. Discrete Math. 313, 26–34 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  2. Albertson, M.O.: The irregularity of a graph. Ars Comb. 46, 219–225 (1997)

    MathSciNet  MATH  Google Scholar 

  3. Chang, G.J., Hu, L., Zhu, X.: Circular choromatic number of Mycielski graphs. Discrete Math. 205, 23–37 (1999)

    Article  MathSciNet  Google Scholar 

  4. Das, K.C.: On geometric arithmetic index of graphs. MATCH Commun. Math. Comput. Chem. 64, 619–630 (2010)

    MathSciNet  MATH  Google Scholar 

  5. Devillers, J., Balaban, A.T. (eds.): Topological Indices and Related Descriptors in QSAR and QSPR. Gordon and Breach, Amsterdam (1999)

  6. Dobrynin, A.A., Kochetova, A.A.: Degree distance of a graph: a degree analogue of the Wiener index. J. Chem. Inf. Comput. Sci. 34, 1082–1086 (1994)

    Article  Google Scholar 

  7. Estrada, E., Torres, L., Rodriguez, L., Gutman, I.: An atom-bond connectivity index: modelling the enthalpy of formation of alkanes. Indian J. Chem. 37A, 849–855 (1998)

    Google Scholar 

  8. Estrada, E.: Atom-bond connectivity and the energetic of branced alkanes. Chem. Phys. Lett. 463, 422–425 (2008)

    Article  Google Scholar 

  9. Furtula, B., Gutman, I., Tomovic, Z., Vesel, A., Pesek, I.: Wiener-type topological indices of phenylenes. Indian J. Chem. 41A, 1767–1772 (2002)

    Google Scholar 

  10. Fisher, D.C., Mckena, P.A., Boyer, E.D.: Hamiltonicity, diameter, domination, packing and biclique partitions of the Mycielskis graphs. Discrete Appl. Math. 84, 93–105 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  11. Gutman, I.: Selected properties of the Schultz molecular topological index. J. Chem. Inf. Comput. Sci. 34, 1087–1089 (1994)

    Article  Google Scholar 

  12. Gutman, I., Polansky, O.E.: Mathematical Concepts in Organic Chemistry. Springer, Berlin (1986)

    Book  MATH  Google Scholar 

  13. Gutman, I.: A property of the Wiener number and its modifications. Indian J. Chem. 36A, 128–132 (1997)

    Google Scholar 

  14. Gutman, I., Dobrynin, A.A., Klavzar, S., Pavlovic, L.: Wiener-type invariants of trees and their relation. Bull. Inst. Combin. Appl. 40, 23–30 (2004)

    MathSciNet  MATH  Google Scholar 

  15. Gutman, I., Vidovic, D., Popovic, L.: Graph representation of organic molecules. Cayley’s plerograms vs. his kenograms. J. Chem. Soc. Faraday Trans. 94, 857–860 (1998)

    Article  Google Scholar 

  16. Hamzeh, A., Iranmanesh, A., Hossein-Zadeh, S., Diudea, M.V.: Generalized degree distance of trees, unicyclic and bicyclic graphs. Stud. Ubb Chem. LVII 4, 73–85 (2012)

    Google Scholar 

  17. Hamzeh, A., Iranmanesh, A., Hossein-Zadeh, S.: Some results on generalized degree distance. Open J. Discrete Math. 3, 143–150 (2013)

    Article  MATH  Google Scholar 

  18. Hua, H., Zhang, S.: On the reciprocal degree distance of graphs. Discrete Appl. Math. 160, 1152–1163 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  19. Liu, H.: Circular choromatic number and Mycielski construction. J. Graph Theory 44, 106–115 (2003)

    Article  MathSciNet  Google Scholar 

  20. Liu, D.D.F.: Circular choromatic number for iterated Mycielski graphs. Discrete Math. 285, 335–340 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  21. Li, S.C., Meng, X.: Four edge-grafting theorems on the reciprocal degree distance of graphs and their applications. J. Comb. Optim. 30, 468–488 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  22. Luo, W., Zhou, B.: On the irregularity of trees and unicyclic graphs with given matching number. Util. Math. 83, 141–147 (2010)

    MathSciNet  MATH  Google Scholar 

  23. Mycielski, J.: Sur le colouring des graphes. Colloq. Math. 3, 161–162 (1955)

    MathSciNet  MATH  Google Scholar 

  24. Palacios, J.L.: Aresistive upper bound for the \(ABC\) index. MATCH Commun. Math. Comput. Chem. 72, 709–713 (2014)

    MathSciNet  MATH  Google Scholar 

  25. Pattabiraman, K., Vijayaragavan, M.: Reciprocal degree distance of product graphs. Discrete Appl. Math. 179, 201–213 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  26. Pattabiraman, K., Kandan, P.: Generalized degree distance of strong product of graphs. Iran. J. Math. Sci. Inf. 10, 87–98 (2015)

    MathSciNet  MATH  Google Scholar 

  27. Pattabiraman, K., Kandan, P.: Generalization on the degree distance of tensor product of graphs. Aust. J. Comb. 62, 211–227 (2015)

    MathSciNet  MATH  Google Scholar 

  28. Su, G.F., Xiong, L.M., Su, X.F., Chen, X.L.: Some results on the reciprocal sum-degree distance of graphs. J. Comb. Optim. 30, 435–446 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  29. Veylaki, M., Nikmehr, M.J., Tavallaee, H.A.: The third and hyper Zagren coindices of some graph operations. J. Appl. Math. Comput. 50, 315–325 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  30. Vukicevic, D., Furtula, B.: Topological index based on the ratios of geometrical and arithmetical means of end-vertex degrees of edges. J. Math. Chem. 46, 1369–1376 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  31. Xing, R., Zhou, B., Du, Z.: Further results on atom-bond connectivity index of trees. Discrete Appl. Math. 158, 1536–1545 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  32. Zhu, H.Y., Klenin, D.J., Lukovits, I.: Extensions of the Wiener number. J. Chem. Inf. Comput. Sci. 36, 420–428 (1996)

    Article  Google Scholar 

  33. Zhou, B., Xing, R.: On atom-bond connectivity index. Z. Naturforsch 66a, 61–66 (2011)

    Article  Google Scholar 

  34. Zhou, B., Luo, W.: On irregularity of graphs. Ars Comb. 88, 55–64 (2008)

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Annamalai University, Annamalainagar, Chidambaram, 608 002, India

    K. Pattabiraman

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  1. K. Pattabiraman
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Pattabiraman, K. On Topological Indices of Graph Transformation. Int. J. Appl. Comput. Math 3, 1171–1184 (2017). https://doi.org/10.1007/s40819-016-0167-6

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