Extremality of graph entropy based on degrees of uniform hypergraphs with few edges

TL;DR

This paper investigates the extremality of a degree-based graph entropy measure for uniform hypergraphs with few edges, deriving bounds for specific hypergraph classes.

Contribution

It provides new bounds on degree-based graph entropy for uniform hypergraphs with few edges, focusing on supertrees, unicyclic, and bicyclic structures.

Findings

Derived upper and lower bounds for entropy in supertrees.

Established bounds for unicyclic uniform hypergraphs.

Analyzed entropy extremality in bicyclic hypergraphs.

Abstract

Let $\mathcal{H}$ be a hypergraph with $n$ vertices. Suppose that $d_1,d_2,\ldots,d_n$ 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.