The article 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 a large number of cospectral digraphs; for every 0≤α≤3, all digraphs whose spectrum is contained in the interval (−α,α) are determined.

We consider the representation of a continuous-time quantum walk in a graph X by the matrix exp(−itA(X)). We provide necessary and sufficient criteria for distance-regular graphs and, more generally, for graphs in association schemes to have perfect state transfer. Using these conditions, we provide several new examples of perfect state transfer in simple graphs.

We study the transition matrix of a quantum walk on strongly regular graphs. It is proposed by Emms, Hancock, Severini and Wilson in 2006, that the spectrum of S + (U 3 ), a matrix based on the amplitudes of walks in the quantum walk, distinguishes strongly regular graphs. We find the eigenvalues of S + (U ) and S + (U 2 ) for regular graphs. Eigenvalues AbstractWe study the transition matrix of a quantum walk on strongly regular graphs. It is proposed by Emms, Hancock, Severini and Wilson in 2006, that the spectrum of S + (U 3 ), a matrix based on the amplitudes of walks in the quantum walk, distinguishes strongly regular graphs. We find the eigenvalues of S + (U ) and S + (U 2 ) for regular graphs and show that S + (U 2 ) = S + (U ) 2 + I.

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In this paper, we show that, for any odd prime p and positive integer t, the path on 2tp−1 vertices admits pretty good state transfer between vertices a and (n + 1 − a) for each a that is a multiple of 2t−1 with respect to the quantum walk model determined by the XY-Hamiltonian. This gives the first examples of pretty good state transfer occurring between internal vertices on an unweighted path, when it does not occur on the extremal vertices.