On Non-localization of Eigenvectors of High Girth Graphs

TL;DR

This paper improves bounds on how localized eigenvectors of high girth regular graphs can be, showing they are more delocalized than previously thought, with near-sharp bounds demonstrated through probabilistic constructions.

Contribution

It provides tighter bounds on eigenvector localization in high girth graphs and introduces a probabilistic construction for near-sharp examples.

Findings

Improved upper bound on eigenvector localization to rac{}{} dlog_d(k)/

Construction of graphs matching the lower bound  ext{Omega}( ext{log}_d(k)/)

Demonstration that eigenvector delocalization bounds are close to optimal

Abstract

We prove improved bounds on how localized an eigenvector of a high girth regular graph can be, and present examples showing that these bounds are close to sharp. This study was initiated by Brooks and Lindenstrauss (2009) who relied on the observation that certain suitably normalized averaging operators on high girth graphs are hyper-contractive and can be used to approximate projectors onto the eigenspaces of such graphs. Informally, their delocalization result in the contrapositive states that for any $\varepsilon \in (0,1)$ and positive integer $k,$ if a $(d+1)-$regular graph has an eigenvector which supports $\varepsilon$ fraction of the $\ell_2^2$ mass on a subset of $k$ vertices, then the graph must have a cycle of size $\tilde{O}(\log_{d}(k)/\varepsilon^2)$, suppressing logarithmic terms in $1/\varepsilon$. In this paper, we improve the upper bound to $\tilde{O}(\log_{d}(k)/\varepsilon)$ and present a construction showing a lower bound of $\Omega(\log_d(k)/\varepsilon)$. Our construction is probabilistic and involves gluing together a pair of trees while maintaining high girth as well as control on the eigenvectors and could be of independent interest.