Hyper-regular graphs and high dimensional expanders

Abstract

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 0for 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.

Near coverings and cosystolic expansion

“…In this work we also analyze partite tensor products of simplicial complexes as defined in [FI20]. We show that a partite tensor product of a k-partite β-coboundary expander with a complete k-partite complex is also a β • exp(−O(k))-coboundary expander.…”

“…and then pairing them by color. This operation on simplicial complexes was defined by [FI20] which also observed that a link of a (k…”