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Acta Mathematica Sinica, English Series

, Volume 35, Issue 7, pp 1238–1250 | Cite as

Extremality of Graph Entropy Based on Degrees of Uniform Hypergraphs with Few Edges

  • Dan HuEmail author
  • Xue Liang LiEmail author
  • Xiao Gang LiuEmail author
  • Sheng Gui ZhangEmail author
Article
  • 11 Downloads

Abstract

Let \(\mathcal{H}\) be a hypergraph with n vertices. Suppose that d1,d2,…,dn are degrees of the vertices of \(\mathcal{H}\). The t-th graph entropy based on degrees of\(\mathcal{H}\) is defined as \(I_{d}^{t}(\mathcal{H})=-\sum\limits_{i=1}^{n}\left(\frac{d_{i}^{t}}{\sum\nolimits_{j=1}^{n}d_{j}^{t}}\log \frac{d_{i}^{t}}{\sum\nolimits_{j=1}^{n}d_{j}^{t}}\right)=\log\left(\sum\limits_{i=1}^{n}d_{i}^{t}\right)-\sum\limits_{i=1}^{n}\left(\frac{d_{i}^{t}}{\sum\nolimits_{j=1}^{n}d_{j}^{t}}\log d_{i}^{t}\right),\) where t is a real number and the logarithm is taken to the base two. In this paper we obtain upper and lower bounds of \(I_{d}^{t}(\mathcal{H})\) for t = 1, when \(\mathcal{H}\) is among all uniform supertrees, unicyclic uniform hypergraphs and bicyclic uniform hypergraphs, respectively.

Keywords

Shannon’s entropy graph entropy degree sequence hypergraph 

MR(2010) Subject Classification

05C50 15A18 

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Notes

Acknowledgements

We greatly appreciate the anonymous referees for their comments and suggestions.

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Copyright information

© Springer-Verlag GmbH Germany & The Editorial Office of AMS 2019

Authors and Affiliations

  1. 1.Department of Applied MathematicsNorthwestern Polytechnical UniversityXi’an, ShaanxiP. R. China
  2. 2.Center for CombinatoricsNankai UniversityTianjinP. R. China
  3. 3.Xi’an-Budapest Joint Research Center for CombinatoricsNorthwestern Polytechnical UniversityXi’an, ShaanxiP. R. China

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