Note on von Neumann and R\'enyi entropies of a Graph

TL;DR

This paper investigates the von Neumann and Rényi entropies of graphs, conjecturing minimal entropy properties for connected graphs and providing proofs for almost all cases, with implications for graph entropy optimization.

Contribution

It introduces a conjecture on the minimal von Neumann entropy for connected graphs and proves it for almost all graphs, also analyzing Rényi entropies and their extremal graphs.

Findings

Connected graphs of order n have von Neumann entropy at least as great as the star graph K_{1,n-1}.

Connected graphs of order n have Rényi 2-entropy at least as great as the star graph.

Complete graphs maximize Rényi α-entropy for α>1 among graphs of order n.

Abstract

We conjecture that all connected graphs of order $n$ have von Neumann entropy at least as great as the star $K_{1,n-1}$ and prove this for almost all graphs of order $n$. We show that connected graphs of order $n$ have R\'enyi 2-entropy at least as great as $K_{1,n-1}$ and for $\alpha>1$, $K_n$ maximizes R\'enyi $\alpha$-entropy over graphs of order $n$. We show that adding an edge to a graph can lower its von Neumann entropy.