For every prime d and any alpha below 1/6, there are infinitely many (d+1)-regular graphs with near-optimal second eigenvalue, logarithmic girth, and many fully localized eigenvectors on sets of size O(m^alpha).
Quantum ergodicity on large regular graphs
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High-girth near-Ramanujan graphs with localized eigenvectors
For every prime d and any alpha below 1/6, there are infinitely many (d+1)-regular graphs with near-optimal second eigenvalue, logarithmic girth, and many fully localized eigenvectors on sets of size O(m^alpha).