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In this paper we discuss potentially practical ways to construct expandergraphs with good spectral properties and a compact description. We considervariations of a constructions that is simple to implement in practice, anddevelop techniques that seem to be applicable to graphs of feasible size. Morespecifically, we focus on expander graphs defined as random Schreier graphs ofthe general linear group over the finite field of size two. We performnumerical experiments and observe that such constructions produce with highprobability Ramanujan graphs that can be useful for practical applications.  To find a theoretical explanation of the observed experimental results andprove an upper bound for the expected second largest eigenvalue of the sampledgraphs, we use the method of moments. We focus on the settings for which itseems difficult to study the asymptotic behaviour of large graphs but it ispossible 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 ofexpander graphs that are, so to speak, pseudo-random (i.e., each graph can beefficiently reconstructed from a short random seed); this approach takes anintermediate position between explicit (deterministic) constructions and theconventional theory of random graphs. Second, we adjust and optimisetheoretical bounds not for the limiting behaviour of graphs but for the valuesof parameters that become meaningful in practical applications (when the wholegraph or at least the indices of its vertices can be stored in computermemory).

Spectral bound for random Schreier graphs of the general linear group