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：本文探讨了生成具有良好谱性质且描述紧凑的展开图的潜在实用方法。我们重点关注几类均匀和二分展开图，这些图定义为大小为二的有限域上一般线性群的随机施莱尔图。通过数值实验，我们表明此类构造产生的谱展开图可应用于实际场景。为从理论上解释观察到的实验结果，我们运用矩方法证明了构造中所用随机施莱尔图的期望第二大特征值的上界。我们关注的是那些难以研究渐近行为、但能对数值实验中参数规模相对较小的图（例如顶点数少于2^200、度数至少为顶点数的对数）计算出非平凡结论的界。