Expander graphs, due to their good mixing properties, are useful in many algorithms and combinatorial constructions. One can produce an expander graph with high probability by taking a random graph. For example, for the case of bipartite graphs of degree $d$ and $n$ vertices in each part we may take independently $d$ permutations of an $n$-element set and use them for edges. 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 reasonably large graphs. The natural idea is to replace the group of all permutations by its small subgroup. Let $n$ be $q^k-1$ for some $k$ and some prime $q$. Then we may interpret vertices as non-zero $k$-dimensional vector over the field $\mathbb{F}_q$, and take random \emph{linear} permutations, i.e., random elements of $GL_k(\mathbb{F}_q)$. In this way the number of random bits used will be polynomial in $k$ (i.e., the logarithm of the graph size, instead of the graph size itself) and the degree. In this paper we provide some experimental data that show that indeed this replacement does not change much the mixing properties (the second eigenvalue) of the random graph that we obtain. These data are provided for several types of graphs (undirected regular and biregular bipartite graphs). We also prove some upper bounds for the second eigenvalue (though it is 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 of graphs only marginally worse, while the number of random bits decreases significantly.

翻译：扩展图因其良好的混合性质，在许多算法和组合构造中非常有用。通过随机图可以以高概率生成扩展图。例如，对于两侧各含 $n$ 个顶点、度数为 $d$ 的二部图，我们可以独立选取 $n$ 元素集合上的 $d$ 个置换，并将其用于边的构造。这种构造比所有已知的显式扩展图构造简单得多，且能以高概率得到具有良好混合性质（较小的第二最大特征值）的图。然而从实践角度看，该构造使用了过多的随机比特，因此对于规模较大的图而言，生成和存储这些比特较为困难。一个自然的想法是将全体置换群替换为其小子群。设 $n = q^k - 1$，其中 $k$ 为正整数，$q$ 为素数。此时可将顶点解释为域 $\mathbb{F}_q$ 上的非零 $k$ 维向量，并选取随机\emph{线性}置换，即 $GL_k(\mathbb{F}_q)$ 中的随机元素。这样，使用的随机比特数将是 $k$（即图规模的对数，而非图规模本身）与度数的多项式函数。本文提供实验数据表明，这种替换对所得随机图的混合性质（第二特征值）影响甚微。这些数据涵盖多种图类型（无向正则图和双正则二部图）。我们还证明了第二特征值的某些上界（尽管与实验结果相比相当弱）。最后，我们讨论了使用Toeplitz矩阵进一步减少随机比特数的可能性；实验表明，这一改变仅使图的混合性质略有恶化，而随机比特数则显著降低。