Abstract
The harmonic index of a graph G is denoted by H(G) and is defined as \(H(G)=\sum _{uv\in E(G)} \frac{2}{d_{u}+d_{v}}\), where \(d_u\), \(d_v\) denote the degrees of the vertices u, v, respectively, of G and E(G) is the edge set of G. In this paper, the graphs having sixth to fifteenth maximum harmonic indices are characterized from the class of all n-vertex trees for sufficiently large n.
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Ali, A.: Counter examples to a conjecture concerning harmonic index. Asian Eur. J. Math. 11(2), 1850035 (2018)
Ali, A., Zhong, L., Gutman, I.: Harmonic index and its generalizations: extremal results and bounds. MATCH Commun. Math. Comput. Chem. 81, 249–311 (2019)
Bondy, J.A., Murty, U.S.R.: Graph Theory. Springer, Berlin (2008)
Chang, R., Zhu, Y.: On the harmonic index and the minimum degree of a graph. Roman. J. Inform. Sci. Technol. 15, 335–343 (2012)
Deng, H., Balachandran, S., Ayyaswamy, S.K., Venkatakrishnan, Y.B.: On harmonic indices of trees, unicyclic graphs and bicyclic graphs. Ars Comb. 130, 239–248 (2017)
Deng, H., Balachandran, S., Balachandar, S.R.: The minimum value of the harmonic index for a graph with the minimum degree two. Asian Eur. J. Math. (2018). https://doi.org/10.1142/S1793557120500540
Fajtlowicz, S.: On conjectures of Graffiti—II. Congr. Numer. 60, 187–197 (1987)
Harary, F.: Graph Theory. Addison-Wesley, Boston (1969)
Hu, Y.M., Zhou, X.Y.: On the harmonic index of the unicyclic and bicyclic graphs. WSEAS Trans. Math. 12, 716–726 (2013)
Ilić, A.: Note on the harmonic index of a graph. Ars Comb. 128, 295–299 (2016)
Jerline, J.A., Michaelraj, L.B., Dhanalakshmi, K., Syamala, P.: Harmonic index of graphs with more than one cut-vertex. Ars Comb. 135, 283–298 (2017)
Li, J., Shiu, W.C.: The harmonic index of a graph. Rocky Mt. J. Math. 44, 1607–1620 (2014)
Liang, M., Cheng, B., Liu, J.: Solution to the minimum harmonic index of graphs with given minimum degree. Trans. Comb. 7, 25–33 (2018)
Lv, J.B.: On the harmonic index of quasi-tree graphs. Ars Comb. 137, 305–315 (2018)
Lv, J., Li, J.: The harmonic index of bicyclic graphs with given matching number. Util. Math. 107, 1–16 (2018)
Martínez-Pérez, Á., Rodríguez, J.M.: Some results on lower bounds for topological indices. J. Math. Chem. 57(5), 1472–1495 (2019)
Matejić, M., Milovanović, I.Ž., Milovanović, E.I.: On bounds for harmonic topological index. Filomat 32, 311–317 (2018)
Ramane, H.S., Basavanagoud, B., Jummannaver, R.B.: Harmonic index and Randić index of generalized transformation graphs. J. Niger. Math. Soc. 37(2), 57–69 (2018)
Ramane, H.S., Joshi, V.B., Jummannaver, R.B., Shindhe, S.D.: Relationship between Randić index, sum-connectivity index, harmonic index and \(\pi \)-electron energy for benzenoid hydrocarbons. Natl. Acad. Sci. Lett. (2019). https://doi.org/10.1007/s40009-019-0782-y. (in press)
Rasi, R., Sheikholeslami, S.M.: The smallest harmonic index of trees with given maximum degree. Discuss. Math. Graph Theory 38, 499–513 (2018)
Sun, X., Gao, Y., Du, J., Xu, L.: On a conjecture of the harmonic index and the minimum degree of graphs. Filomat 32, 3435–3441 (2018)
Zhong, L.: The harmonic index for graphs. Appl. Math. Lett. 25, 561–566 (2012)
Acknowledgements
The authors would immensely grateful to the anonymous referees for their valuable comments and useful suggestions. The third author’s research is supported by University of Haifa, Israel, for the Postdoctoral studies and it is gratefully acknowledged.
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Ali, A., Balachandran, S., Elumalai, S. et al. On the n-vertex trees with sixth to fifteenth maximum harmonic indices. Afr. Mat. 31, 771–780 (2020). https://doi.org/10.1007/s13370-019-00758-0
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DOI: https://doi.org/10.1007/s13370-019-00758-0