  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.
