More on minors of Hermitian (quasi-)Laplacian matrix of the second kind for mixed graphs

A mixed graph $M_{G}$ is the graph obtained from an unoriented simple graph $G$ by giving directions to some edges of $G$, where $G$ is often called the underlying graph of $M_{G}$. In this paper, we introduce two classes of incidence matrices of the second kind of $M_{G}$, and discuss the determinants of these two matrices for rootless mixed trees and unicyclic mixed graphs. Applying these results, we characterize the explicit expressions of various minors for Hermitian (quasi-)Laplacian matrix of the second kind of $M_{G}$. Moreover, we give two sufficient conditions that the absolute values of all the cofactors of Hermitian (quasi-)Laplacian matrix of the second kind are equal to the number of spanning trees of the underlying graph $G$.Comment: 16 pages,7 figure

On spectra of Hermitian Randic matrix of second kind

We propose the Hermitian Randi\'c matrix $R^\omega(X)=(R^\omega_{ij})$, where $\omega=\frac{1+i \sqrt{3}}{2}$ and $R^\omega_{ij}={1}/{\sqrt{d_id_j}}$ if $v_iv_j$ is an unoriented edge, ${\omega}/{\sqrt{d_id_j}}$ if $v_i\rightarrow v_j$, ${\overline{\omega}}/{\sqrt{d_id_j}}$ if $v_i\leftarrow v_j$, and 0 otherwise. This appears to be more natural because of $\omega+\overline{\omega}=1$ and $|\omega|=1$. In this paper, we investigate some 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