On the largest eigenvalue of a mixed graph with partial orientation

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