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

Let H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{H}$$\end{document} be a hypergraph with n vertices. Suppose that d1,d2,…,dn are degrees of the vertices of H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{H}$$\end{document}. The t-th graph entropy based on degrees ofH\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{H}$$\end{document} is defined as Idt(H)=−∑i=1n(dit∑j=1ndjtlogdit∑j=1ndjt)=log(∑i=1ndit)−∑i=1n(dit∑j=1ndjtlogdit),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I_{d}^{t}(\mathcal{H})=-\sum\limits_{i=1}^{n}\left(\frac{d_{i}^{t}}{\sum\nolimits_{j=1}^{n}d_{j}^{t}}\log \frac{d_{i}^{t}}{\sum\nolimits_{j=1}^{n}d_{j}^{t}}\right)=\log\left(\sum\limits_{i=1}^{n}d_{i}^{t}\right)-\sum\limits_{i=1}^{n}\left(\frac{d_{i}^{t}}{\sum\nolimits_{j=1}^{n}d_{j}^{t}}\log d_{i}^{t}\right),$$\end{document} 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 Idt(H)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I_{d}^{t}(\mathcal{H})$$\end{document} for t = 1, when H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{H}$$\end{document} is among all uniform supertrees, unicyclic uniform hypergraphs and bicyclic uniform hypergraphs, respectively.