Krystal Guo

Hermitian Adjacency Matrix of Digraphs and Mixed Graphs

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.

Perfect state transfer on distance-regular graphs and association schemes

Abstract We consider the representation of a continuous-time quantum walk in a graph X by the matrix exp ⁡ ( − i t A ( 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.

New constructions of complete non-cyclic hadamard matrices, related function families and LCZ sequences

A Hadamard matrix is said to be completely non-cyclic (CNC) if there are no two rows (or two columns) that are shift equivalent in its reduced form. In this paper, we present three new constructions of CNC Hadamard matrices. We give a primary construction using a flipping operation on the submatrices of the reduced form of a Hadamard matrix. We show that, up to some restrictions, the Kronecker product preserves the CNC property of Hadamard matrices and use this fact to give two secondary constructions of Hadamard matrices. The applications to construct low correlation zone sequences are provided.

Spectral Bound for Separations in Eulerian Digraphs

The spectra of digraphs, unlike those of graphs, is a relatively unexplored territory. In a digraph, a separation is a pair of sets of vertices X and Y such that there are no arcs from X and Y . For a subclass of eulerian digraphs, we give an bound on the size of a separation in terms of the eigenvalues of the Laplacian matrix.

Digraphs with Hermitian spectral radius below 2 and their cospectrality with paths

Abstract It is well-known that the paths are determined by the spectrum of the adjacency matrix. For digraphs, every digraph whose underlying graph is a tree is cospectral to its underlying graph with respect to the Hermitian adjacency matrix ( H -cospectral). Thus every (simple) digraph whose underlying graph is isomorphic to P n is H -cospectral to P n . Interestingly, there are others. This paper finds digraphs that are H -cospectral with the path graph P n and whose underlying graphs are nonisomorphic, when n is odd, and finds that such graphs do not exist when n is even. In order to prove this result, all digraphs whose Hermitian spectral radius is smaller than 2 are determined.