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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Alizadeh, Y., Klavžar, S.: Complexity of topological indices: the case of connective eccentric index. MATCH Commun. Math. Comput. Chem. 76, 659–667 (2016)
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)
Aouchiche, M., Hansen, P.: Proximity and remoteness in graphs: results and conjectures. Networks 58, 95–102 (2011)
Bera, S., Mahalingam, K.: Structural properties of word representable graphs. Math. Comp. Sci. 10, 209–222 (2016)
Bondy, J.A., Murty, U.S.R.: Graph Theory with Applications. Macmillan London and Elsevier, New York (1976)
Chung, F.R.K.: The average distance and the independence number. J. Graph Theory 12, 229–235 (1988)
Collins, A., Kitaev, S., Lozin, V.: New results on word-representable graphs. Discrete Appl. Math. 216, 136–141 (2017)
Das, K.C., Gutman, I.: On Wiener and multiplicative Wiener indices of graphs. Discrete Appl. Math. 206, 9–14 (2016)
Diestel, R.: Graph Theory. Springer, Berlin (2006)
Dobrynin, A., Entringer, R., Gutman, I.: Wiener index of trees: theory and applications. Acta Appl. Math. 66, 211–249 (2001)
Fajtlowicz, S., Waller, W.A.: On two conjectures of GRAFFITI II. Congr. Numer. 60, 187–197 (1987)
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)
Gutman, I., Trinajstić, N.: Graph theory and molecular orbitals. Total \(\pi \)-electron energy of alternant hydrocarbons, Chem. Phys. Lett. 17 (1972) 535-538
Halldórsson, M.M., Kitaev, S., Pyatkin, A.: Graphs capturing alternations in words, Lecture Notes Comput. Sci., Proc. DLT 6224, 436-437 (2010)
Hua, H., Das, K.C.: The relationship between the eccentric connectivity index and Zagreb indices. Discrete Appl. Math. 161, 2480–2491 (2013)
Hua, H., Das, K.C.: Proof of conjectures on remoteness and proximity in graphs. Discrete Appl. Math. 171, 72–80 (2014)
Hua, H., Chen, Y., Das, KCh.: The difference between remoteness and radius of a graph. Discrete Appl. Math. 187, 103–110 (2015)
Hua, H., Wang, H., Hu, X.: On eccentric distance sum and degree distance of graphs. Discrete Appl. Math. 250, 262–275 (2018)
Hua, H., Das, K.C.: Comparative results and bounds for the eccentric-adjacency index. Discrete Appl. Math. 285, 188–196 (2020)
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)
Ilić, A., Yu, G., Feng, L.: On the eccentric distance sum of graphs. J. Math. Anal. Appl. 381, 590–600 (2011)
Kitaev, S., Lozin, V.: Words and Graphs, vol. 17. Springer (2015)
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)
Masre, M., Vetrík, T.: General degree-eccentricity index of trees, Bull. Malays. Math. Sci. Soc. (in press)
Mateescu, A., Salomaa, A., Salomaa, K., Yu, S.: A sharpening of the Parikh mapping. RAIRO Theor. Inform. Appl. 35(6), 551–564 (2001)
Lothaire, M.: Combinatorics on Words Cambridge Mathematical Library, Cambridge university Press, (1997)
Nikolić, S., Kovačević, G., Miličević, A., Trinajstić, N.: The Zagreb indices 30 years after. Croat. Chem. Acta 76, 113–124 (2003)
Rozenberg, G., Salomaa, A.: Handbook of Formal Languages. Springer (1997)
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
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)
Wang, H., Hua, H., Wang, M.: Comparative study of distance-based graph invariants. J. Appl. Math. Comput. 64, 457–469 (2020)
Wiener, H.: Structural determination of paraffin boiling point. J. Amer. Chem. Soc. 69, 17–20 (1947)
Wu, B., An, X., Liu, G., Yan, G., Liu, X.: Minimum degree, edge-connectivity and radius. J. Combin. Optim. 26, 585–591 (2013)
Xu, K., Alizadeh, Y., Das, K.: On two eccentricity-based topological indices of graphs, Discrete Appl. Math. 233, 240–251 (2017)
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)
Yu, G., Feng, L.: On the connective eccentricity index of graphs. MATCH Commun. Math. Comput. Chem. 69, 611–628 (2013)
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)
Zhang, L., Wu, B.: The Nordhaus-Gaddum-type inequalities for some chemical indices. MATCH Commun. Math. Comput. Chem. 54, 189–194 (2005)
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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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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DOI: https://doi.org/10.1007/s40840-021-01140-9