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Zagreb Indices and Multiplicative Zagreb Indices of Eulerian Graphs

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Abstract

For a graph \(G = (V(G), E(G))\), let d(u), d(v) be the degrees of the vertices u, v in G. The first and second Zagreb indices of G are defined as \( M_1(G) = \sum _{u \in V(G)} d(u)^2\) and \( M_2(G) = \sum _{uv \in E(G)} d(u)d(v)\), respectively. The first (generalized) and second Multiplicative Zagreb indices of G are defined as \(\Pi _{1,c}(G) = \prod _{v \in V(G)}d(v)^c\) and \(\Pi _2(G) = \Pi _{uv \in E(G)} d(u)d(v)\), respectively. The (Multiplicative) Zagreb indices have been the focus of considerable research in computational chemistry dating back to Narumi and Katayama in 1980s. Denote by \({\mathcal {G}}_{n}\) the set of all Eulerian graphs of order n. In this paper, we characterize Eulerian graphs with first three smallest and largest Zagreb indices and Multiplicative Zagreb indices in \({\mathcal {G}}_{n}\).

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Acknowledgements

This work is partially supported by National Natural Science Foundation of China (Nos. 11601006, 11471016, 11401004, 11571134, 11371162), Anhui Provincial Natural Science Foundation (Nos. KJ2015A331, KJ2013B105, 1408085QA03), the Self-determined Research Funds of Central China Normal University from the colleges basic research and operation of MOE. The authors would like to express their sincere gratitude to the anonymous referees and the editor for many friendly and helpful suggestions, which led to great deal of improvement of the original manuscript.

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Authors and Affiliations

  1. School of Mathematics and Physics, Anhui Jianzhu University, Hefei, 230601, People’s Republic of China

    Jia-Bao Liu

  2. School of Mathematics and Statistics, Central China Normal University, Wuhan, 430079, People’s Republic of China

    Chunxiang Wang

  3. Department of Mathematics and Computer Science, Adelphi University, Garden City, NY, 11530, USA

    Shaohui Wang

  4. Department of Mathematics, The University of Mississippi, University, MS, 38677, USA

    Bing Wei

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  1. Jia-Bao Liu
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  2. Chunxiang Wang
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  3. Shaohui Wang
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Correspondence to Shaohui Wang.

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Communicated by Xueliang Li.

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Liu, JB., Wang, C., Wang, S. et al. Zagreb Indices and Multiplicative Zagreb Indices of Eulerian Graphs. Bull. Malays. Math. Sci. Soc. 42, 67–78 (2019). https://doi.org/10.1007/s40840-017-0463-2

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  • DOI: https://doi.org/10.1007/s40840-017-0463-2

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