Hermitian-Randi\'c matrix and Hermitian-Randi\'c energy of mixed graphs

Let $M$ be a mixed graph and $H(M)$ be its Hermitian-adjacency matrix. If we add every edge and arc in $M$ a Randi'c weight, then we can get a new weighted Hermitian-adjacency matrix. What are the properties of this new matrix? Motivated by this, we define the Hermitian-Randi'c matrix $R_H(M)=(r_h{)}_{kl}$ of a mixed graph $M$, where $(r_h{)}_{kl}=-(r_h{)}_{lk}=\frac{\mathbf{\text{i}}}{\sqrt{d_k{d}_l}}$ ($\mathbf{\text{i}}=\sqrt{-1}$) if $(v_k,v_l)$ is an arc of $M$, $(r_h{)}_{kl}=(r_h{)}_{lk}=\frac{1}{\sqrt{d_k{d}_l}}$ if $v_k{v}_l$ is an undirected edge of $M$, and $(r_h{)}_{kl}=0$ otherwise. In this paper, firstly, we compute the characteristic polynomial of the Hermitian-Randi'c matrix of a mixed graph. Furthermore, we give bounds to the Hermitian-Randi'c energy of a general mixed graph. Finally, we give some results about the Hermitian-Randi'c energy of mixed trees.