In this paper we discuss potentially practical ways to produce expander graphs with good spectral properties and a compact description. We focus on several classes of uniform and bipartite expander graphs defined as random Schreier graphs of the general linear group over the finite field of size two. We perform numerical experiments and show that such constructions produce spectral expanders that can be useful for practical applications. To find a theoretical explanation of the observed experimental results, we used the method of moments to prove upper bounds for the expected second largest eigenvalue of the random Schreier graphs used in our constructions. We focus on bounds for which it is difficult to study the asymptotic behaviour but it is possible to compute non-trivial conclusions for relatively small graphs with parameters from our numerical experiments (e.g., with less than 2^200 vertices and degree at least logarithmic in the number of vertices).

翻译：本文讨论了生成具有良好谱性质及紧凑描述的扩展器图的潜在实用方法。我们重点研究几类定义为大小为2的有限域上一般线性群随机施赖尔图的均匀和二分扩展器图。通过数值实验，我们证明此类构造产生的谱扩展器可适用于实际应用。为解释实验观测结果的理论机制，我们采用矩方法证明了构造中所用随机施赖尔图期望第二大特征值的上界。我们重点关注那些难以研究渐近行为，但能基于数值实验参数（例如顶点数少于2^200且度至少为顶点数的对数）对相对较小的图计算非平凡结论的边界。