On spectra of Hermitian Randić matrix of second kind

Let $X$ be a mixed graph and $\omega =\frac{1+\mathrm{ı}\sqrt{3}}{2}$. We write $i\to j$, if there is an oriented edge from a vertex $v_i$ to another vertex $v_j$, and $i\sim j$ for an un-oriented edge between the vertices $v_i$ and $v_j$. The degree of a vertex $v_i$ is denoted by $d_i$. We propose the Hermitian Randić matrix of second kind $R^{\text{omeg}}(X)(R_{ij}^{\text{omeg}})$, where $R_{ij}^{\text{omeg}}=\frac{1}{\sqrt{d_i{d}_j}}$ if $i\sim j$, $R_{ij}^{\text{omeg}}=\frac{\omega }{\sqrt{d_i{d}_j}}$ and $R_{ji}^{\text{omeg}}=\frac{\bar{\omega }}{\sqrt{d_i{d}_j}}$ if $i\to j$, and 0 otherwise. In this paper, we investigate some spectral features of this novel Hermitian matrix and study a few properties like positiveness, bipartiteness, edge-interlacing etc. We also compute the characteristic polynomial for this new matrix and obtain some upper and lower bounds for the eigenvalues and the energy of this matrix.