Hyper-regular graphs and high dimensional expanders

Abstract

Let G = (V, E) be a finite graph. For d₀ > 0 we say that G is d₀-regular, if every v ∈ V has degree d₀. We say that G is (d₀, d₁)-regular, for 0 < d₁ < d₀, if G is d₀ regular and for every v ∈ V, the subgraph induced on v’s neighbors is d₁-regular. Similarly, G is (d₀, d₁,⋯,d_n−1)-regular for 0 < d_n−1 < ⋯ < d₁ < d₀, if G is d₀ regular and for every v ∈ V, the subgraph induced on v’s neighbors is (d₁,⋯,d_n−1)-regular (i.e., for every 1 ≤ i ≤ n − 1, the joint neighborhood of every clique of size i is d_i-regular); in that case, we say that G is an n-dimensional hyper-regular graph (HRG). Here we define a new kind of graph product, through which we build examples of infinite families of n-dimensional HRG such that the joint neighborhood of every clique of size at most n − 1 is connected. In particular, relying on the work of Kaufman and Oppenheim, our product yields an infinite family of n-dimensional HRG for arbitrarily large n with good expansion properties. This answers a question of Dinur regarding the existence of such objects.