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 show 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：本文讨论构造具有良好谱性质和紧凑描述的扩张图的实际可行方法。我们考虑一种在实际中易于实现的构造变体，并发展出适用于可行规模图的技术。具体而言，我们关注基于大小为2的有限域上一般线性群的随机Schreier图定义的扩张图。通过数值实验，我们证明此类构造能以高概率生成对实际应用有用的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概率特征值的期望上界，我们采用矩方法。我们聚焦于难以研究大规模图渐进行为但能为实际应用所需的较小规模图提供非平凡界的设定。本文的主要贡献有两方面：首先，我们研究所谓的伪随机扩张图族（即每张图可由短随机种子高效重构），该方法介于显式（确定性）构造与传统随机图理论之间；其次，我们将理论界调整并优化至适用于实际应用中具有意义的参数值（当整张图或至少其顶点索引可存储于计算机内存时）。