On the principal eigenvector of a graph

TL;DR

This paper investigates how the principal eigenvector ratio of a graph, which measures irregularity, changes under small modifications, revealing bounds based on spectral gaps and graph structure.

Contribution

It provides bounds on the stability of the principal ratio under edge modifications, linking spectral gaps to irregularity measures.

Findings

Removing an edge in a cycle of bounded length keeps the ratio polynomially bounded.

Joining vertices at distance two can cause the ratio to jump exponentially.

A large additive spectral gap ensures the ratio remains bounded after edge modifications.

Abstract

The principal ratio of a connected graph $G$, $\gamma(G)$, is the ratio between the largest and smallest coordinates of the principal eigenvector of the adjacency matrix of $G$. Over all connected graphs on $n$ vertices, $\gamma(G)$ ranges from $1$ to $n^{cn}$. Moreover, $\gamma(G)=1$ if and only if $G$ is regular. This indicates that $\gamma(G)$ can be viewed as an irregularity measure of $G$, as first suggested by Tait and Tobin (El. J. Lin. Alg. 2018). We are interested in how stable this measure is. In particular, we ask how $\gamma$ changes when there is a small modification to a regular graph $G$. We show that this ratio is polynomially bounded if we remove an edge belonging to a cycle of bounded length in $G$, while the ratio can jump from $1$ to exponential if we join a pair of vertices at distance $2$. We study the connection between the spectral gap of a regular graph and the stability of its principal ratio. A naive bound shows that given a constant multiplicative spectral gap and bounded degree, the ratio remains polynomially bounded if we add or delete an edge. Using results from matrix perturbation theory, we show that given an additive spectral gap greater than $(2+\epsilon)\sqrt{n}$, the ratio stays bounded after adding or deleting an edge.