Let $V$ be a set of $n$ vertices. Fix $3 \le k \le n$. Let $\binom V k$ be the set of all $k$ element subsets of $V$.

We add the edges in $V$ as follows: Let $\mathcal S \subseteq \binom V k$ be fixed. For each $F \in \mathcal S$, I am making the vertices in $F$ mutually adjacent. Let's call this graph $G_k(\mathcal S)$.

1. I want to learn how the graph $G_k(\mathcal S)$ looks like?

2. Is there any name for $G_k(\mathcal S)$ in the literature?

3. Some references regarding these graphs.

Kindly share your thoughts. Thank you.