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On the ABC index of connected graphs with given degree sequences

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Abstract

The atom-bond connectivity (ABC) index of a graph G is defined to be \(ABC(G)=\sum _{uv\in E(G)}\sqrt{\frac{d(u)+d(v)-2}{d(u)d(v)}}\) where d(u) is the degree of a vertex u. The ABC index plays a key role in correlating the physical–chemical properties and the molecular structures of some families of compounds. In this paper, we describe the structural properties of graphs which have the minimum ABC index among all connected graphs with a given degree sequence. Moreover, these results are used to characterize the extremal graphs which have the minimum ABC index among all unicyclic and bicyclic graphs with a given degree sequence.

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Authors and Affiliations

  1. Department of Mathematics, Shanghai Sanda University, 2727 Jinhai Road, Shanghai, 201209, People’s Republic of China

    Xiu-Mei Zhang

  2. Department of Mathematics, Shanghai University of Electric Power, 2103 Changyang Road, Shanghai, 200090, People’s Republic of China

    Yu-Qin Sun

  3. Department of Mathematical Sciences, Georgia Southern University, Statesboro, GA, 30460, USA

    Hua Wang

  4. School of Mathematical Science, and MOE-LSC, Shanghai Jiao Tong University, 800 Dongchuan Road, Shanghai, 200240, People’s Republic of China

    Xiao-Dong Zhang

Authors
  1. Xiu-Mei Zhang
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  2. Yu-Qin Sun
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  3. Hua Wang
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  4. Xiao-Dong Zhang
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Corresponding author

Correspondence to Xiao-Dong Zhang.

Additional information

This work is supported by the National Natural Science Foundation of China (Nos. 11531001, 11271256, 11701372, 61702291 and 11401368), the Joint NSFC-ISF Research Program, jointly funded by the National Natural Science Foundation of China and the Israel Science Foundation (Grant No. 11561141001), Innovation Program of Shanghai Municipal Education Commission (Nos. 14ZZ016, 15ZZ108), Shanghai Natural Science Foundation of China (No. 16ZR1422400), the Scientific Research Starting Foundation for High-level Talents of Pingdingshan University (No. PXY-BSQD2017006), and the Simons Foundation (No. 245307).

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Zhang, XM., Sun, YQ., Wang, H. et al. On the ABC index of connected graphs with given degree sequences. J Math Chem 56, 568–582 (2018). https://doi.org/10.1007/s10910-017-0802-4

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  • DOI: https://doi.org/10.1007/s10910-017-0802-4

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