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 .

中文翻译：

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在正则图上保留频谱的短周期去除

我们描述了一种新方法，可以在保持谱边界（邻接矩阵的非平凡特征值）的同时去除常规图上的短循环，只要图具有某些组合属性。这些组合属性与短循环之间的数量和距离有关，并且在均匀随机的正则图中以很高的概率发生。使用这种方法，我们可以显示两个涉及高周长频谱扩展器图的结果。首先，我们证明给定 $d\geq 3$ 和 $n$，存在 $d$-正则 $\Theta(n)$-顶点图的显式分布，其中很有可能其样本具有周长 $\Omega( \log_{d - 1} n)$ 并且是 $\epsilon$-near-Ramanujan; 即，它的特征值的大小以 $2\sqrt{d - 1} + \epsilon$ 为界（不包括 $d$ 的单个平凡特征值）。然后，