Extremal entropy for graphs with given size

TL;DR

This paper characterizes the extremal graphs with a fixed number of edges that minimize the first degree-based entropy, identifying colex graphs as the minimizers, thereby clarifying the measure's interpretation.

Contribution

It proves that colex graphs minimize the first degree-based entropy among graphs with a given size, establishing a key extremal property.

Findings

Colex graphs minimize the first degree-based entropy for fixed size

Provides a rigorous proof of extremal entropy properties

Clarifies the interpretation of degree-based entropy in graph theory

Abstract

The first degree-based entropy of a graph is the Shannon entropy of its degree sequence normalized by the degree sum. Its correct interpretation as a measure of uniformity of the degree sequence requires the determination of its extremal values given natural constraints. In this paper, we prove that the graphs with given size that minimize the first degree-based entropy are the colex graphs.