  It is proved that for any finite connected graph $G$, there exists an
orientation of $G$ such that the spectral radius of the corresponding Hermitian
adjacency matrix is smaller or equal to the spectral radius of the universal
cover of $G$ (with equality if and only if $G$ is a tree). This resolves a
problem proposed by Mohar. The proof uses the method of interlacing families of
polynomials that was developed by Marcus, Spielman, and Srivastava in their
seminal work on the existence of infinite families of Ramanujan graphs.
