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

Abstract

Let $\mathcal{H}$ be a hypergraph with n vertices. Suppose that d₁,d₂,…,d_n 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.