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 $d_1,d_2,\dots ,d_n$ are degrees of the vertices of $\mathcal{H}$. The $t$-th graph entropy based on degrees of $\mathcal{H}$ is defined as $$I_d^t(\mathcal{H})=-\sum _{i=1}^n\left(\frac{d_i^t}{{\sum }_{j=1}^n{d}_j^t}\log \frac{d_i^t}{{\sum }_{j=1}^n{d}_j^t}\right)=\log \left(\sum _{i=1}^n{d}_i^t\right)-\sum _{i=1}^n\left(\frac{d_i^t}{{\sum }_{j=1}^n{d}_j^t}\log d_i^t\right),$$ 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.