  We prove that for each $d \geq 3$ the set of all limit points of the second
largest eigenvalue of growing sequences of $d$-regular graphs is
$[2\sqrt{d-1},d]$. A similar argument shows that the set of all limit points of
the smallest eigenvalue of growing sequences of $d$-regular graphs with growing
(odd) girth is $[-d, -2 \sqrt{d-1}]$. The more general question of identifying
all vectors which are limit points of the vectors of the top $k$ eigenvalues of
sequences of $d$-regular graphs is considered as well. As a by product, in the
study of discrete counterpart of the "scarring" phenomenon observed in the
investigation of quantum ergodicity on manifolds, our technique provides a
method to construct $d$-regular almost Ramanujan graphs with large girth and
localized eigenvectors corresponding to eigenvalues larger than $2\sqrt{d-1}$,
strengthening a result of Alon, Ganguly, and Srivastava.
