Main eigenvalues for normal matrices and mixed graphs

In this work, we deal with mixed graphs G^ and their associated non-negative mixed adjacency matrices, that is, matrices whose entries are one whenever there is a link between the vertices, and zero otherwise. Then, normal mixed graphs, mixed graphs whose non-negative mixed adjacency matrix is normal, are presented and main eigenvalues (eigenvalues which have an associated eigenvector not orthogonal to the all ones vector) are studied. Normal mixed graphs with exactly one or exactly two main eigenvalues are discussed. Main eigenvalues of some normal matrices in block superdiagonal form are studied. For an undirected simple graph H, the main eigenvalues of the mixed H-join of normal mixed regular graphs are discussed. Moreover, weighted mixed graphs and normal weighted mixed graphs are defined and, as consequence, if H^ is a normal weighted mixed graph, the normal H^-weighted join mixed graph is constructed. Finally, families of normal H^-weighted join of mixed graphs with an odd number of main eigenvalues are constructed.