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-

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    $\begingroup$ This is essentially the nerve of the path groupoid of your graph. $\endgroup$ – Zhen Lin May 21 '14 at 19:22

Self-answer: This is the clique complex.


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