Spectral bound for random Schreier graphs of the general linear group

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).