Example 1.1.6.4. To every simplicial set $X$, we can associate a directed graph $\mathrm{Gr}( X)$ as follows:
The vertex set $\operatorname{Vert}( \mathrm{Gr}(X) )$ is the set of $0$-simplices of the simplicial set $X$.
The edge set $\operatorname{Edge}( \mathrm{Gr}(X) )$ is the set of nondegenerate $1$-simplices of the simplicial set $X$.
For every edge $e \in \operatorname{Edge}( \mathrm{Gr}(X) )$, the source $s(e)$ is the vertex $d^{1}_1(e)$, and the target $t(e)$ is the vertex $d^{1}_0(e)$ (here $d^{1}_0$ and $d^{1}_1$ denote the face operators of Construction 1.1.1.4).