Exercise 1, § 6

For a graph G to be the graph of an equivalence relation on a set E, it is necessary and sufficient that $\mathrm{pr_1G} = \mathrm{E}$, $\mathrm{G} \circ \overset{-1}{\mathrm{G}} \circ \mathrm{G} = \mathrm{G}$, and $\Delta_{\mathrm{E}} \subset \mathrm{G}$ ($\Delta_{\mathrm{E}}$ being the diagonal of E).