Erdős-Rényi Sequences and Deterministic Construction of Expanding Cayley Graphs

Given a finite group G by its multiplication table as input, we give a deterministic polynomial-time construction of a directed Cayley graph on G with O(log |G|) generators, which has a rapid mixing property and a constant spectral expansion. We prove a similar result in the undirected case, and give a new deterministic polynomialtime construction of an expanding Cayley graph with O(log |G|) generators, for any group G given by its multiplication table. This gives a completely different and elementary proof of a result of Wigderson and Xiao [10]. For any finite group G given by a multiplication table, we give a deterministic polynomialtime construction of a cube generating sequence that gives a distribution onG which is arbitrarily close to the uniform distribution. This derandomizes the well-known construction of Erdös-Rényi sequences [2].