Hermitian adjacency matrix of digraphs and mixed graphs

Abstract: The paper gives a thorough introduction to spectra of digraphs via its Hermitian adjacency matrix. This matrix is indexed by the vertices of the digraph, and the entry corresponding to an arc from $x$ to $y$ is equal to the complex unity $i$ (and its symmetric entry is $-i$) if the reverse arc $yx$ is not present. We also allow arcs in both directions and unoriented edges, in which case we use $1$ as the entry. This allows to use the definition also for mixed graphs. This matrix has many nice properties; it has real eigenvalues and the interlacing theorem holds for a digraph and its induced subdigraphs. Besides covering the basic properties, we discuss many differences from the properties of eigenvalues of undirected graphs and develop basic theory. The main novel results include the following. Several surprising facts are discovered about the spectral radius; some consequences of the interlacing property are obtained; operations that preserve the spectrum are discussed -- they give rise to an incredible number of cospectral digraphs; for every $0\le\alpha\le\sqrt{3}$, all digraphs whose spectrum is contained in the interval $(-\alpha,\alpha)$ are determined.