Hyper-regular graphs and high dimensional expanders

Let G = (V, E) be a finite graph. For d ₀ > 0 we say that G is d ₀-regular, if every v ∈ V has degree d ₀. We say that G is (d ₀, d ₁)-regular, for 0 < d ₁ < d ₀, if G is d ₀ regular and for every v ∈ V, the subgraph induced on v’s neighbors is d ₁-regular. Similarly, G is (d ₀, d ₁,⋯,d_n−1)-regular for 0 < d_n−1 < ⋯ < d ₁ < d ₀, if G is d ₀ regular and for every v ∈ V, the subgraph induced on v’s neighbors is (d ₁,⋯,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.