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The connective eccentricity index and modified second Zagreb index of Parikh word representable graphs

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Abstract

The connective eccentricity index is a degree-distance-based graph invariant, while the modified second Zagreb index is a degree-based graph invariant. The Parikh word representable graph is a new class of graphs G(w), which corresponds to words w that are finite sequence of symbols. In this paper, we first present explicit formulas for the connective eccentricity index and modified second Zagreb index of the Parikh word representable graphs corresponding to binary core words of the form aub over a binary alphabet \(\{a,\,b\}\), respectively. Then, we investigate the relationship between the connective eccentricity index and modified second Zagreb index and obtain some comparative results about these two indices.

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References

  1. Alizadeh, Y., Klavžar, S.: Complexity of topological indices: the case of connective eccentric index. MATCH Commun. Math. Comput. Chem. 76, 659–667 (2016)

    MathSciNet  MATH  Google Scholar 

  2. Aouchiche, M., Bonnefoy, J.M., Fidahoussen, A., Caporossi, G., Hansen, P., Hiesse, L., Lachere, J., Monhait, A.: Variable neighborhood search for extremal graphs. 14. The autoGraphiX 2 system. In: Liberti, L., Maculan, N. (eds.) Global Optimization: From Theory to Implementation, pp. 281–310. Springer, New York (2006)

  3. Aouchiche, M., Hansen, P.: Proximity and remoteness in graphs: results and conjectures. Networks 58, 95–102 (2011)

    Article  MathSciNet  Google Scholar 

  4. Bera, S., Mahalingam, K.: Structural properties of word representable graphs. Math. Comp. Sci. 10, 209–222 (2016)

    Article  MathSciNet  Google Scholar 

  5. Bondy, J.A., Murty, U.S.R.: Graph Theory with Applications. Macmillan London and Elsevier, New York (1976)

    Book  Google Scholar 

  6. Chung, F.R.K.: The average distance and the independence number. J. Graph Theory 12, 229–235 (1988)

    Article  MathSciNet  Google Scholar 

  7. Collins, A., Kitaev, S., Lozin, V.: New results on word-representable graphs. Discrete Appl. Math. 216, 136–141 (2017)

    Article  MathSciNet  Google Scholar 

  8. Das, K.C., Gutman, I.: On Wiener and multiplicative Wiener indices of graphs. Discrete Appl. Math. 206, 9–14 (2016)

    Article  MathSciNet  Google Scholar 

  9. Diestel, R.: Graph Theory. Springer, Berlin (2006)

    MATH  Google Scholar 

  10. Dobrynin, A., Entringer, R., Gutman, I.: Wiener index of trees: theory and applications. Acta Appl. Math. 66, 211–249 (2001)

    Article  MathSciNet  Google Scholar 

  11. Fajtlowicz, S., Waller, W.A.: On two conjectures of GRAFFITI II. Congr. Numer. 60, 187–197 (1987)

    MATH  Google Scholar 

  12. Gupta, S., Singh, M., Madan, A.K.: Connective eccentricity index: a novel topological descriptor for predicting biological activity. J. Mol. Graph. Model. 18, 18–25 (2000)

    Article  Google Scholar 

  13. Gutman, I., Trinajstić, N.: Graph theory and molecular orbitals. Total \(\pi \)-electron energy of alternant hydrocarbons, Chem. Phys. Lett. 17 (1972) 535-538

  14. Halldórsson, M.M., Kitaev, S., Pyatkin, A.: Graphs capturing alternations in words, Lecture Notes Comput. Sci., Proc. DLT 6224, 436-437 (2010)

  15. Hua, H., Das, K.C.: The relationship between the eccentric connectivity index and Zagreb indices. Discrete Appl. Math. 161, 2480–2491 (2013)

    Article  MathSciNet  Google Scholar 

  16. Hua, H., Das, K.C.: Proof of conjectures on remoteness and proximity in graphs. Discrete Appl. Math. 171, 72–80 (2014)

    Article  MathSciNet  Google Scholar 

  17. Hua, H., Chen, Y., Das, KCh.: The difference between remoteness and radius of a graph. Discrete Appl. Math. 187, 103–110 (2015)

  18. Hua, H., Wang, H., Hu, X.: On eccentric distance sum and degree distance of graphs. Discrete Appl. Math. 250, 262–275 (2018)

    Article  MathSciNet  Google Scholar 

  19. Hua, H., Das, K.C.: Comparative results and bounds for the eccentric-adjacency index. Discrete Appl. Math. 285, 188–196 (2020)

    Article  MathSciNet  Google Scholar 

  20. Hua, H.: On the quotients between the eccentric connectivity index and the eccentric distance sum of graphs with diameter 2. Discrete Appl. Math. 285, 297–300 (2020)

    Article  MathSciNet  Google Scholar 

  21. Ilić, A., Yu, G., Feng, L.: On the eccentric distance sum of graphs. J. Math. Anal. Appl. 381, 590–600 (2011)

    Article  MathSciNet  Google Scholar 

  22. Kitaev, S., Lozin, V.: Words and Graphs, vol. 17. Springer (2015)

  23. Klavžar, S., Nadjafi-Arani, M.J.: Improved bounds on the difference between the Szeged index and the Wiener index of graphs. European J. Combin. 39, 148–156 (2014)

    Article  MathSciNet  Google Scholar 

  24. Masre, M., Vetrík, T.: General degree-eccentricity index of trees, Bull. Malays. Math. Sci. Soc. (in press)

  25. Mateescu, A., Salomaa, A., Salomaa, K., Yu, S.: A sharpening of the Parikh mapping. RAIRO Theor. Inform. Appl. 35(6), 551–564 (2001)

    Article  MathSciNet  Google Scholar 

  26. Lothaire, M.: Combinatorics on Words Cambridge Mathematical Library, Cambridge university Press, (1997)

  27. Nikolić, S., Kovačević, G., Miličević, A., Trinajstić, N.: The Zagreb indices 30 years after. Croat. Chem. Acta 76, 113–124 (2003)

    Google Scholar 

  28. Rozenberg, G., Salomaa, A.: Handbook of Formal Languages. Springer (1997)

  29. Thomas, N., Mathew, L., Sriram, S., Subramanian, K.G.: Wiener-type indices of Parikh word representable graphs, Ars Math. Contemp., Available online at: https://doi.org/10.26493/1855-3974.2359.a7b

  30. Vukičević, D., Trinajstić, N.: Modified Zagreb \(M_2\) index-Comparison with the Randić connectivity index for benzenoid systems. Croatica Chem. Acta 76, 183–187 (2003)

    Google Scholar 

  31. Wang, H., Hua, H., Wang, M.: Comparative study of distance-based graph invariants. J. Appl. Math. Comput. 64, 457–469 (2020)

    Article  MathSciNet  Google Scholar 

  32. Wiener, H.: Structural determination of paraffin boiling point. J. Amer. Chem. Soc. 69, 17–20 (1947)

    Article  Google Scholar 

  33. Wu, B., An, X., Liu, G., Yan, G., Liu, X.: Minimum degree, edge-connectivity and radius. J. Combin. Optim. 26, 585–591 (2013)

    Article  MathSciNet  Google Scholar 

  34. Xu, K., Alizadeh, Y., Das, K.: On two eccentricity-based topological indices of graphs, Discrete Appl. Math. 233, 240–251 (2017)

  35. Xu, K., Das, K.C., Liu, H.: Some extremal results on the connective eccentricity index of graphs. J. Math. Anal. Appl. 433, 803–817 (2016)

    Article  MathSciNet  Google Scholar 

  36. Yu, G., Feng, L.: On the connective eccentricity index of graphs. MATCH Commun. Math. Comput. Chem. 69, 611–628 (2013)

    MathSciNet  MATH  Google Scholar 

  37. Yu, G., Qu, H., Tang, L., Feng, L.: On the connective eccentricity index of trees and unicyclic graphs with given diameter. J. Math. Anal. Appl. 420, 1776–1786 (2014)

    Article  MathSciNet  Google Scholar 

  38. Zhang, L., Wu, B.: The Nordhaus-Gaddum-type inequalities for some chemical indices. MATCH Commun. Math. Comput. Chem. 54, 189–194 (2005)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors are grateful to two anonymous referees for their detailed suggestions which have considerably improved the presentation of the paper. This research was supported by National Natural Science Foundation of China under Grant Nos. 11971011, 11571135 and sponsored by Qing Lan Project of Jiangsu Province, P.R. China.

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Correspondence to Hongbo Hua.

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Communicated by Sanming Zhou.

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Hua, H., Wang, M. The connective eccentricity index and modified second Zagreb index of Parikh word representable graphs. Bull. Malays. Math. Sci. Soc. 44, 3689–3704 (2021). https://doi.org/10.1007/s40840-021-01140-9

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