Revisiting the Maximum Principal Ratio of Graphs

TL;DR

This paper proves that for all sufficiently large graphs with at least 5000 vertices, the kite graph uniquely maximizes the principal ratio of the Perron eigenvector among all connected graphs.

Contribution

It confirms the conjecture that the kite graph has the maximum principal ratio for all connected graphs with at least 5000 vertices.

Findings

Kite graph attains maximum principal ratio for n ≥ 5000.

Conjecture by Cioabă and Gregory is validated for all large n.

The result extends previous confirmation for large n to all n ≥ 5000.

Abstract

Let $G$ be a connected graph, the principal ratio of $G$ is the ratio of the maximum and minimum entries of its Perron eigenvector. In 2007, Cioab\v a and Gregory conjectured that among all connected graphs on $n$ vertices, the kite graph attains the maximum principal ratio. In 2018, Tait and Tobin confirmed the conjecture for sufficientlty large $n$. In this article, we show the conjecture is true for all $n\geq 5000$.