Characterizing graphs of maximum principal ratio

TL;DR

This paper proves a conjecture that among all connected graphs on n vertices, the kite graph maximizes the principal ratio, which is the ratio of the maximum to minimum entries of the first eigenvector.

Contribution

The paper confirms the conjecture that the kite graph uniquely maximizes the principal ratio among connected graphs, advancing understanding of eigenvector properties.

Findings

Kite graph maximizes principal ratio for connected graphs

Proof of the conjecture confirming the kite graph's extremal property

Enhances understanding of eigenvector ratios in graph theory

Abstract

The principal ratio of a connected graph, denoted $\gamma(G)$, is the ratio of the maximum and minimum entries of its first eigenvector. Cioab\u{a} and Gregory conjectured that the graph on $n$ vertices maximizing $\gamma(G)$ is a kite graph: a complete graph with a pendant path. In this paper we prove their conjecture.