Expander graphs, due to their mixing properties, are useful in many algorithms and combinatorial constructions. One can produce an expander graph with high probability by taking a random graph (e.g., the union of $d$ random bijections for a bipartite graph of degree $d$). This construction is much simpler than all known explicit constructions of expanders and gives graphs with good mixing properties (small second largest eigenvalue) with high probability. However, from the practical viewpoint, it uses too many random bits, so it is difficult to generate and store these bits for large graphs. The natural idea is to restrict the class of the bijections that we use. For example, if both sides are linear spaces $\mathbb{F}_q^k$ over a finite field $\mathbb{F}_q$, we may consider only \emph{linear} bijections, making the number of random bits polynomial in $k$ (and not $q^k$). In this paper we provide some experimental data that shows that this approach conserves the mixing properties (the second eigenvalue) for several types of graphs (undirected regular and biregular bipartite graphs). We also prove some upper bounds for the second eigenvalue (though they are quite weak compared with the experimental results). Finally, we discuss the possibility to decrease the number of random bits further by using Toeplitz matrices; our experiments show that this change makes the mixing properties only marginally worse while the number of random bits decreases significantly.

翻译：扩展图因其混合特性在许多算法和组合构造中具有重要应用。通过随机图（例如，对于度为$d$的二部图，取$d$个随机双射的并集）可以高概率生成扩展图。该构造方法比所有已知的显式扩展图构造更为简单，并能以高概率获得具有良好混合特性（次大特征值较小）的图。然而，从实际应用的角度来看，该方法需要消耗过多随机比特，因此对于大规模图而言，生成和存储这些随机比特较为困难。一种自然的思路是限制所用双射的类别。例如，若图的两侧均为有限域$\mathbb{F}_q$上的线性空间$\mathbb{F}_q^k$，则可仅考虑\emph{线性}双射，使得随机比特数变为$k$的多项式级别（而非$q^k$量级）。本文通过实验数据表明，该方法在多种图类型（无向正则图及二部双正则图）中能保持混合特性（次大特征值）。同时，我们证明了次大特征值的若干上界（尽管这些理论界与实验结果相比仍显宽松）。最后，我们探讨了通过使用Toeplitz矩阵进一步减少随机比特数的可能性；实验表明，这种改进仅会轻微降低混合特性，同时能显著减少随机比特的消耗量。