General Degree-Eccentricity Index of Trees

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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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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Communicated by Sanming Zhou.

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

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Keywords

  • General degree-eccentricity index
  • Tree
  • Matching number
  • Independence number
  • Domination number
  • Eccentric connectivity index

Mathematics Subject Classification

  • 05C05
  • 92E10
  • 05C12