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On the n-vertex trees with sixth to fifteenth maximum harmonic indices

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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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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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Author notes

  1. Akbar Ali

    Present address: Department of Mathematics, Faculty of Science, University of Ha’il, Ha’il, Saudi Arabia

  2. Selvaraj Balachandran

    Present address: Department of Mathematics and Applied Mathematics, University of the Free State, Bloemfontein, South Africa

Authors and Affiliations

  1. Knowledge Unit of Science, University of Management and Technology, Sialkot, Pakistan

    Akbar Ali

  2. Department of Mathematics, School of Arts, Sciences and Humanities, SASTRA Deemed University, Thanjavur, India

    Selvaraj Balachandran

  3. Department of Mathematics, University of Haifa, 3498838, Haifa, Israel

    Suresh Elumalai & Toufik Mansour

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  1. Akbar Ali
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  2. Selvaraj Balachandran
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  3. Suresh Elumalai
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  4. Toufik Mansour
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Correspondence to Akbar Ali.

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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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