Resolution of Yan's conjecture on entropy of graphs

TL;DR

This paper characterizes connected graphs with fixed order and size that minimize the first degree-based entropy, extending Yan's conjecture and providing a complete solution for certain graph parameters.

Contribution

It proves Yan's conjecture by identifying the connected graphs with minimal degree-based entropy for specified order and size ranges.

Findings

Connected graphs with minimal entropy are characterized for given parameters.

The conjecture by Yan is extended and proved for the range $n-1 \,\le\, m \le 2n-3$.

The results provide insights into the entropy properties of graphs based on degree sequences.

Abstract

The first degree-based entropy of a graph is the Shannon entropy of its degree sequence normalized by the degree sum. In this paper, we characterize the connected graphs with given order $n$ and size $m$ that minimize the first degree-based entropy whenever $n-1 \le m \le 2n-3,$ thus extending and proving a conjecture by Yan.