Hermitian adjacency matrix of the second kind for mixed graphs

This contribution gives an extensive study on spectra of mixed graphs via its Hermitian adjacency matrix of the second kind introduced by Mohar [21]. This matrix is indexed by the vertices of the mixed graph, and the entry corresponding to an arc from $u$ to $v$ is equal to the sixth root of unity $omega=frac{1+{bf i}sqrt{3}}{2}$ (and its symmetric entry is $overline{omega}=frac{1-{bf i}sqrt{3}}{2}$); the entry corresponding to an undirected edge is equal to 1, and 0 otherwise. The main results of this paper include the following: Some interesting properties are discovered about the characteristic polynomial of this novel matrix. Cospectral problems among mixed graphs, including mixed graphs and their underlying graphs, are studied. We give equivalent conditions for a mixed graph that shares the same spectrum of its Hermitian adjacency matrix of the second kind ($H_S$-spectrum for short) with its underlying graph. A sharp upper bound on the $H_S$-spectral radius is established and the corresponding extremal mixed graphs are identified. Operations which are called three-way switchings are discussed--they give rise to a large number of $H_S$-cospectral mixed graphs. We extract all the mixed graphs whose rank of its Hermitian adjacency matrix of the second kind ($H_S$-rank for short) is $2$ (resp. 3). Furthermore, we show that all connected mixed graphs with $H_S$-rank $2$ can be determined by their $H_S$-spectrum. However, this does not hold for all connected mixed graphs with $H_S$-rank $3$. We identify all mixed graphs whose eigenvalues of its Hermitian adjacency matrix of the second kind ($H_S$-eigenvalues for short) lie in the range $(-alpha,, alpha)$ for $alphainleft{sqrt{2},,sqrt{3},,2right}$.