Explicit Unique-Neighbor Expanders

We present a simple, explicit construction of an infinite family F of bounded-degree ’unique-neighbor’ expanders Γ; i.e., there are strictly positive constants α and , such that all Γ = (X,E(Γ)) ∈ F satisfy the following property. For each subset S of X with no more than α|X| vertices, there are at least |S| vertices in X \ S that are adjacent in Γ to exactly one vertex in S. The construction of F is simple to specify, and each Γ ∈ F is 6-regular. We then extend the technique and present easy to describe explicit infinite families of 4-regular and 3-regular unique-neighbor expanders, as well as explicit families of bipartite graphs with non equal color classes and similar properties. This has several applications and settles an open problem considered by various researchers.