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On the extremal graphs with respect to the total reciprocal edge-eccentricity

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Abstract

The total reciprocal edge-eccentricity of a graph G is defined as \(\xi ^{ee}(G)=\sum _{u\in V_G}\frac{d_G(u)}{\varepsilon _G(u)}\), where \(d_G(u)\) is the degree of u and \(\varepsilon _G(u)\) is the eccentricity of u. In this paper, we first characterize the unique graph with the maximum total reciprocal edge-eccentricity among all graphs with a given number of cut vertices. Then we determine the k-connected bipartite graphs of order n with diameter d having the maximum total reciprocal edge-eccentricity. Finally, we identify the unique tree with the minimum total reciprocal edge-eccentricity among the n-vertex trees with given degree sequence.

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Acknowledgements

The authors wish to thank the anonymous referees for their careful reading and valuable comments on how to improve this paper.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Lanzhou University, Lanzhou, 730000, Gansu, People’s Republic of China

    Lifang Zhao & Yuping Gao

  2. Zhongshan Overseas Chinese Secondary School, Zhongshan, 528400, Guangdong, People’s Republic of China

    Hongshuai Li

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  1. Lifang Zhao
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  2. Hongshuai Li
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  3. Yuping Gao
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Correspondence to Yuping Gao.

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Financially supported by the Fundamental Research Funds for the Central Universities (No. lzujbky-2017-28).

Yuping Gao: NSFC(No. 11901263).

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Zhao, L., Li, H. & Gao, Y. On the extremal graphs with respect to the total reciprocal edge-eccentricity. J Comb Optim 39, 115–137 (2020). https://doi.org/10.1007/s10878-019-00458-2

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