Lectures 16 : Constructions of Expanders via the Zig - Zag Graph Product

A family of expanders is a family of graphs Gn = (Vn, En), |Vn| = n, such that each graph is dn-regular, and the edge-expansion of each graph is at least h, for an absolute constant h independent of n. Ideally, we would like to have such a construction for each n, although it is usually enough for most applications that, for some constant c and every k, there is an n for which the construction applies in the interval {k, k + 1, . . . , ck}, or even the interval {k, . . . , ck}. We would also like the degree dn to be slowly growing in n and, ideally, to be bounded above by an explicit constant. Today we will see a simple construction in which dn = O(log 2 n) and a more complicated one in which dn = O(1). An explicit construction of a family of expanders is a construction in which Gn is “efficiently computable” given n. The weakest sense in which a construction is said to be explicit is when, given n, the (adjacency matrix of the) graph Gn can be constructed in time polynomial in n. A stronger requirement, which is necessary for several applications, is that given n and i ∈ {1, . . . , n}, the list of neighbors of the i-th vertex of Gn can be computed in time polynomial in log n.