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Super Edge-connectivity and Zeroth-order General Randić Index for −1 ≤ α < 0

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Abstract

Let G be a connected graph with order n, minimum degree δ = δ(G) and edge-connectivity λ = λ(G). A graph G is maximally edge-connected if λ = δ, and super edge-connected if every minimum edgecut consists of edges incident with a vertex of minimum degree. Define the zeroth-order general Randić index \(R_\alpha ^0\left( G \right) = \sum\limits_{x \in V\left( G \right)} {d_G^\alpha \left( x \right)} \) , where dG(x) denotes the degree of the vertex x. In this paper, we present two sufficient conditions for graphs and triangle-free graphs to be super edge-connected in terms of the zeroth-order general Randić index for −1 ≤ α < 0, respectively.

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Acknowledgements

The authors wish to thank the referee for useful comments and suggestions that have allowed them to improve the presentation of the paper.

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Authors and Affiliations

  1. School of Mathematics and Information Science, Yantai University, Yantai, 264005, China

    Zhi-hong He

  2. Department of Mathematical Science, Tsinghua University, Beijing, 100084, China

    Mei Lu

Authors
  1. Zhi-hong He
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  2. Mei Lu
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Correspondence to Zhi-hong He.

Additional information

This work is supported by the National Natural Science Foundation of China (No.11501490, 61373019, 11371307) and by the Natural Science Foundation of Shandong Province (No. ZR2015AM006).

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He, Zh., Lu, M. Super Edge-connectivity and Zeroth-order General Randić Index for −1 ≤ α < 0. Acta Math. Appl. Sin. Engl. Ser. 34, 659–668 (2018). https://doi.org/10.1007/s10255-018-0775-5

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  • DOI: https://doi.org/10.1007/s10255-018-0775-5

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