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General Degree-Eccentricity Index of Trees

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Abstract

For a connected graph G and \(a,b \in \mathbb {R}\), the general degree-eccentricity index is defined as \(\mathrm{DEI}_{a,b}(G) = \sum _{v \in V(G)} d_{G}^{a}(v) \mathrm{ecc}_{G}^{b}(v)\), where V(G) is the vertex set of G, \(d_{G} (v)\) is the degree of a vertex v and \(\mathrm{ecc}_{G}(v)\) is the eccentricity of v in G. We obtain sharp upper and lower bounds on the general degree-eccentricity index for trees of given order in combination with given matching number, independence number, domination number or bipartition. The bounds hold for \(0< a < 1\) and \(b > 0\), or for \(a > 1\) and \(b < 0\). Many bounds hold also for \(a = 1\). All the extremal graphs are presented.

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Funding

The work of T. Vetrík is based on the research supported by the National Research Foundation of South Africa (Grant No. 129252).

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Authors and Affiliations

  1. Department of Mathematics, Woldia University, Woldia, Ethiopia

    Mesfin Masre

  2. Department of Mathematics and Applied Mathematics, University of the Free State, Bloemfontein, South Africa

    Tomáš Vetrík

Authors
  1. Mesfin Masre
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  2. Tomáš Vetrík
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Mesfin Masre is a PhD student who was working under the supervision of Tomáš Vetrík.

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Correspondence to Tomáš Vetrík.

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The authors declare that they have no conflict of interest.

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Communicated by Sanming Zhou.

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Masre, M., Vetrík, T. General Degree-Eccentricity Index of Trees. Bull. Malays. Math. Sci. Soc. 44, 2753–2772 (2021). https://doi.org/10.1007/s40840-021-01086-y

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  • DOI: https://doi.org/10.1007/s40840-021-01086-y

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