Graphs with minimum degree-based entropy

TL;DR

This paper investigates the extremal properties of degree-based entropy in graphs, identifying the structures that minimize this entropy for given vertices and edges, with implications for trees and bipartite graphs.

Contribution

It characterizes the unique extremal graphs minimizing degree-based entropy among graphs and bipartite graphs with fixed vertices and edges.

Findings

The star graph minimizes degree-based entropy among trees.

A lower bound for bipartite graphs' degree-based entropy is established.

All extremal bipartite graphs achieving the bound are characterized.

Abstract

The degree-based entropy of a graph is defined as the Shannon entropy based on the information functional that associates the vertices of the graph with the corresponding degrees. In this paper, we study extremal problems of finding the graphs attaining the minimum degree-based graph entropy among graphs and bipartite graphs with a given number of vertices and edges. We characterize the unique extremal graph achieving the minimum value among graphs with a given number of vertices and edges and present a lower bound for the degree-based entropy of bipartite graphs and characterize all the extremal graphs which achieve the lower bound. This implies the known result due to Cao et al. (2014) that the star attains the minimum value of the degree-based entropy among trees with a given number of vertices.