New matrices for the spectral theory of mixed graphs, Part II

The concept of the integrated adjacency matrix for mixed graphs was first introduced in 9, where its spectral properties were analyzed in relation to the structural characteristics of the mixed graph. Building upon this foundation, this paper introduces the integrated Laplacian matrix, the integrated signless Laplacian matrix, and the normalized integrated Laplacian matrix for mixed graphs. We further explore how the spectra of these matrices relate to the structural properties of the mixed graph.