On the characteristic polynomial and energy of Hermitian quasi-Laplacian matrix of mixed graphs

A mixed graph is a graph whose edge set consists of both oriented and unoriented edges. The Hermitian-adjacency matrix of an n-vertex mixed graph is a square matrix H(M) = [h(jk)] of order n, where h(jk) = iota? = -h(kj) if there is an arc from v(j) to v(k) and h(jk) = 1 if there is an edge between v(j) and v(k), and h(jk) = 0 otherwise. Let D(M) = [d(jj)] be a diagonal matrix, where d(jj) is the degree of v(j) in the underlying graph of M. The matrices L(M) = D(M) - H(M) and Q(M) = D(M) + H(M) are, respectively, the Hermitian Laplacian and Hermitian quasi-Laplacian matrix of the mixed graph M. In this paper, we first found coefficients of the characteristic polynomial of Hermitian Laplacian and Hermitian quasi-Laplacian matrices of the mixed graph M. Second, we discussed relationship between the spectra of Hermitian Laplacian and Hermitian quasi-Laplacian matrices of the mixed graph M.