Hermitian-adjacency matrices and Hermitian energies of mixed graphs

A complex adjacency matrix of a mixed graph is introduced in the present paper, which is a Hermitian matrix and called the Hermitian-adjacency matrix. It incorporates both adjacency matrix of an undirected graph and skew-adjacency matrix of an oriented graph. Some of its properties are studied. Furthermore, properties of its characteristic polynomial are studied. Cospectral problems among mixed graphs, including mixed graphs and their underlying graphs, oriented graphs and their underlying graphs, are studied. We give equivalent conditions for a mixed graph (especially oriented graph) that share the same spectrum with its underlying graph. As a consequence, we reconfirm a conjecture which was proposed by Cui and Hou in Ref. [8]. We also show that the spectrum of the Hermitian matrix of a mixed graph is invariant when changing the value of any its cut edge (if any).Correspondingly, an energy of a mixed graph is introduced and called the Hermitian energy. It incorporates both the energy of an undirected graph and the skew energy of an oriented graph. Some of its bounds are given. Especially, the mixed graphs with optimal upper bound of Hermitian energy are characterized. An infinite family of mixed graphs attaining the maximum Hermitian energy is constructed. Moreover, the Hermitian energy of a mixed tree is showed to be equal to the energy of its underlying tree. Finally, the integral formula for Hermitian energy of a mixed graph is given.