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Inequalities involving the harmonic-arithmetic index

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Abstract

Let G be a simple graph with vertex set \(V=\{v_{1},v_{2},\ldots ,v_{n}\}\). The notion \(i\sim j\) is used to indicate that the vertices \(v_{i}\) and \(v_{j}\) of G are adjacent. For a vertex \(v_{i}\in V\), let \(d_{i}\) be the degree of \(v_{i}\). The harmonic-arithmetic (HA) index of G is defined as \(HA(G) =\sum _{i\sim j} 4d_id_j(d_i+d_j)^{-2}\). In this paper, a considerable number of inequalities involving the HA index and other topological indices are derived. For every obtained inequality, all the graphs that satisfy the equality case are also characterized.

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Author notes

  1. Akbar Ali, Emina Milovanović, Stefan Stankov, Marjan Matejić contributed equally to this work.

Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, University of Ha’il, Ha’il, Saudi Arabia

    Akbar Ali

  2. Faculty of Electronic Engineering, University of Niš, Niš, 1800, Serbia

    Emina Milovanović, Stefan Stankov, Marjan Matejić & Igor Milovanović

Authors
  1. Akbar Ali
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  2. Emina Milovanović
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  3. Stefan Stankov
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  4. Marjan Matejić
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  5. Igor Milovanović
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Correspondence to Igor Milovanović.

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Ali, A., Milovanović, E., Stankov, S. et al. Inequalities involving the harmonic-arithmetic index. Afr. Mat. 35, 46 (2024). https://doi.org/10.1007/s13370-024-01183-8

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