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Zagreb Indices and Multiplicative Zagreb Indices of Eulerian Graphs

  • Jia-Bao Liu
  • Chunxiang Wang
  • Shaohui WangEmail author
  • Bing Wei
Article

Abstract

For a graph \(G = (V(G), E(G))\), let d(u), d(v) be the degrees of the vertices uv 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 \({\mathcal {G}}_{n}\).

Keywords

Extremal bounds Zagreb index Multiplicative Zagreb index Eulerian graphs 

Mathematics Subject Classification

05C12 05C05 

Notes

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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Copyright information

© Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2017

Authors and Affiliations

  • Jia-Bao Liu
    • 1
  • Chunxiang Wang
    • 2
  • Shaohui Wang
    • 3
    Email author
  • Bing Wei
    • 4
  1. 1.School of Mathematics and PhysicsAnhui Jianzhu UniversityHefeiPeople’s Republic of China
  2. 2.School of Mathematics and StatisticsCentral China Normal UniversityWuhanPeople’s Republic of China
  3. 3.Department of Mathematics and Computer ScienceAdelphi UniversityGarden CityUSA
  4. 4.Department of MathematicsThe University of MississippiUniversityUSA

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