On spectra of Hermitian Randi\'c matrix of second kind

Abstract

Let $X$ be a mixed graph and $\omega=\frac{1+\i \sqrt{3}}{2}$. We write $i\rightarrow j$, 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 Randi\'c matrix of second kind $R^\omeg(X)\coloneqq(R^\omeg_{ij})$, where $R^\omeg_{ij}=\frac{1}{\sqrt{d_id_j}}$ if $i \sim j$, $R^\omeg_{ij}= \frac{\omega}{\sqrt{d_id_j}}$ and $R^\omeg_{ji}= \frac{\overline{\omega}}{\sqrt{d_id_j}}$ if $i\rightarrow j$, 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.