Acta Mathematica Sinica, English Series

, Volume 35, Issue 7, pp 1238–1250

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

• Dan Hu
• Xue Liang Li
• Xiao Gang Liu
• Sheng Gui Zhang
Article

## 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

05C50 15A18

## Notes

### Acknowledgements

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

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