Hyper-regular graphs and high dimensional expanders

Understand

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).