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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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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DOI: https://doi.org/10.1007/s10878-021-00724-2