Starting with a graph $G$, form a simplicial complex $X$ which has $G$ as the 1-skeleton, and then has higher dimensional simplices whenever more than two vertices of $G$ are mutually adjacent. So any $n$ mutually adjacent vertices in $G$ will correspond to a $(n-1)$-simplex in $X$.
Is this a valid/common construction? Does it have a name? Thanks-
Self-answer: This is the clique complex.