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 $v\in V$ 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 $v\in V$, the subgraph induced on $v$'s neighbors is $d_1$-regular. Similarly, $G$ is $(d_0, d_1,\ldots, d_{n-1})$-regular for $0<d_{n-1}<\ldots<d_1<d_0$, if $G$ is $d_0$ regular and for every $1\leq i\leq n-1$, 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.