On the largest eigenvalue of a mixed graph with partial orientation

Let G be a connected graph and let T be an acyclic set of edges of G. A partial orientation σ of G with respect to T is an orientation of the edges of G except those edges of T, the resulting graph associated with which is denoted by GTσ. In this paper we prove that there exists a partial orientation σ of G with respect to T such that the largest eigenvalue of the Hermitian adjacency matrix of GTσ is at most the largest absolute value of the roots of the matching polynomial of G.