In this paper we discuss potentially practical ways to construct expander graphs with good spectral properties and a compact description. We consider variations of a constructions that is simple to implement in practice, and develop techniques that seem to be applicable to graphs of feasible size. More specifically, we focus on expander graphs defined as random Schreier graphs of the general linear group over the finite field of size two. We perform numerical experiments and observe that such constructions produce with high probability Ramanujan graphs that can be useful for practical applications. To find a theoretical explanation of the observed experimental results and prove an upper bound for the expected second largest eigenvalue of the sampled graphs, we use the method of moments. We focus on the settings for which it seems difficult to study the asymptotic behaviour of large graphs but it is possible to provide non-trivial bounds for graphs of relatively small size (interesting for practical applications). The main contribution of this work is twofold. First, we study families of expander graphs that are, so to speak, pseudo-random (i.e., each graph can be efficiently reconstructed from a short random seed); this approach takes an intermediate position between explicit (deterministic) constructions and the conventional theory of random graphs. Second, we adjust and optimise theoretical bounds not for the limiting behaviour of graphs but for the values of parameters that become meaningful in practical applications (when the whole graph or at least the indices of its vertices can be stored in computer memory).

翻译：本文讨论了构建具有良好谱性质及紧凑描述的扩展图（expander graphs）的潜在实用方法。我们研究了一种易于实际实现的构建方式的变体，并开发了适用于可行规模图的技术。具体而言，我们聚焦于在大小为2的有限域上定义的一般线性群的随机Schreier图作为扩展图。通过数值实验，我们发现此类构造能以高概率产生适用于实际应用的Ramanujan图。为从理论上解释观测到的实验结果，并证明采样图第二大特征值的期望上界，我们采用了矩方法。我们重点关注那些难以研究大图渐近行为、但能对实际应用感兴趣的中等规模图提供非平凡界的情景。本文主要贡献有两方面：其一，我们研究了可称为"伪随机"（即每张图均可从短随机种子高效重构）的扩展图族；该方法介于显式（确定性）构造与经典随机图理论之间。其二，我们将理论界的调整与优化重点从图的极限行为转向实际应用中有意义的参数值（此时整张图或至少顶点索引可存储在计算机内存中）。