A complex square matrix $U$ is unitary iff $U^*U=I$, equivalently iff its columns form an orthonormal basis. Using this graph's convention, $\langle u_i,u_j\rangle=u_i^*u_j=\delta_{ij}$. Since $U$ is square this also gives $UU^*=I$, so its rows are orthonormal as well.
Rectangular matrices can have orthonormal columns without being unitary: then $U^*U=I$ but $UU^*$ is generally a projection. The square hypothesis must not be dropped.
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