  We describe a new method to remove short cycles on regular graphs while
maintaining spectral bounds (the nontrivial eigenvalues of the adjacency
matrix), as long as the graphs have certain combinatorial properties. These
combinatorial properties are related to the number and distance between short
cycles and are known to happen with high probability in uniformly random
regular graphs.
  Using this method we can show two results involving high girth spectral
expander graphs. First, we show that given $d \geq 3$ and $n$, there exists an
explicit distribution of $d$-regular $\Theta(n)$-vertex graphs where with high
probability its samples have girth $\Omega(\log_{d - 1} n)$ and are
$\epsilon$-near-Ramanujan; i.e., its eigenvalues are bounded in magnitude by
$2\sqrt{d - 1} + \epsilon$ (excluding the single trivial eigenvalue of $d$).
Then, for every constant $d \geq 3$ and $\epsilon > 0$, we give a deterministic
poly$(n)$-time algorithm that outputs a $d$-regular graph on
$\Theta(n)$-vertices that is $\epsilon$-near-Ramanujan and has girth
$\Omega(\sqrt{\log n})$, based on the work of arXiv:1909.06988 .
