Average eccentricity, minimum degree and maximum degree in graphs


Let G be a connected finite graph with vertex set V(G). The eccentricity e(v) of a vertex v is the distance from v to a vertex farthest from v. The average eccentricity of G is defined as \(\frac{1}{|V(G)|}\sum _{v \in V(G)}e(v)\). We show that the average eccentricity of a connected graph of order n, minimum degree \(\delta \) and maximum degree \(\Delta \) does not exceed \(\frac{9}{4} \frac{n-\Delta -1}{\delta +1} \big ( 1 + \frac{\Delta -\delta }{3n} \big ) + 7\), and this bound is sharp apart from an additive constant. We give improved bounds for triangle-free graphs and for graphs not containing 4-cycles.

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Correspondence to P. Dankelmann.

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P. Dankelmann: Financial Support by the South African National Research Foundation, Grant Number 118521, is gratefully acknowledged. F.J. Osaye: The results presented in this paper form part of the second author’s PhD thesis.

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Dankelmann, P., Osaye, F.J. Average eccentricity, minimum degree and maximum degree in graphs. J Comb Optim (2020). https://doi.org/10.1007/s10878-020-00616-x

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  • Average eccentricity
  • Distance
  • Minimum degree
  • Maximum degree
  • Eccentricity
  • Graph