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On the relation between Wiener index and eccentricity of a graph

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Abstract

The relation between the Wiener index W(G) and the eccentricity \(\varepsilon (G)\) of a graph G is studied. Lower and upper bounds on W(G) in terms of \(\varepsilon (G)\) are proved and extremal graphs characterized. A Nordhaus–Gaddum type result on W(G) involving \(\varepsilon (G)\) is given. A sharp upper bound on the Wiener index of a tree in terms of its eccentricity is proved. It is shown that in the class of trees of the same order, the difference \(W(T) - \varepsilon (T)\) is minimized on caterpillars. An exact formula for \(W(T) - \varepsilon (T)\) in terms of the radius of a tree T is obtained. A lower bound on the eccentricity of a tree in terms of its radius is also given. Two conjectures are proposed. The first asserts that the difference \(W(G) - \varepsilon (G)\) does not increase after contracting an edge of G. The second conjecture asserts that the difference between the Wiener index of a graph and its eccentricity is largest on paths.

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Acknowledgements

S.K. acknowledges the financial support from the Slovenian Research Agency (research core funding P1-0297 and projects J1-9109, J1-1693, N1-0095, N1-0108). K.C. Das was supported by the National Research Foundation of the Korean government with grant No. 2017R1D1A1B03028642.

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Authors and Affiliations

  1. Department of Mathematics, Esfarayen University of Technology, Esfarayen, Iran

    Hamid Darabi

  2. Department of Mathematics, Hakim Sabzevari University, Sabzevar, Iran

    Yaser Alizadeh

  3. Faculty of Mathematics and Physics, University of Ljubljana, Ljubljana, Slovenia

    Sandi Klavžar

  4. Faculty of Natural Sciences and Mathematics, University of Maribor, Maribor, Slovenia

    Sandi Klavžar

  5. Institute of Mathematics, Physics and Mechanics, Ljubljana, Slovenia

    Sandi Klavžar

  6. Department of Mathematics, Sungkyunkwan University, Suwon, 16419, Republic of Korea

    Kinkar Chandra Das

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  1. Hamid Darabi
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  2. Yaser Alizadeh
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  3. Sandi Klavžar
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  4. Kinkar Chandra Das
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Correspondence to Sandi Klavžar.

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Darabi, H., Alizadeh, Y., Klavžar, S. et al. On the relation between Wiener index and eccentricity of a graph. J Comb Optim 41, 817–829 (2021). https://doi.org/10.1007/s10878-021-00724-2

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