Hermitian Adjacency Matrix of Digraphs and Mixed Graphs

Abstract

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.

A new kind of Hermitian matrices for digraphs

“…which implies that ρ ≤ 3ν 1 . While the factor 1 3 in Theorem 3.2 is tight for Hermitian matrices of the first kind [7], it is not tight for the second kind, where it can be strengthened as follows. Proof.…”

Hermitian Laplacian matrix and positive of mixed graphs

“…Firstly we need some concepts which were introduced in [1] and [2]. The switching operation introduced by Seidel [8] plays an important role in discussions of signed graphs.…”

“…This matrix was introduced by Liu and Li [1] and independently by Guo and Mohar [2]. We will denote by λ i (M) the jth largest eigenvalue of H(M) (multiplicities counted).…”

“…Correspondingly they introduced the Hermitian energy of mixed graph and derived some bounds for Hermitian energy, characterized the mixed graph with these optimal bounds. Guo and Mohar [2] investigated the spectral radius of mixed graphs and the interlacing properties for Hermitian adjacency spectrum. Moreover, they present some operations which preserve the Hermitian spectrum of mixed graphs.…”

Discrete-Time Quantum Walks on Oriented Graphs

“…and the resulting evolution operator is U = U 2 U 1 . From the viewpoint of graph theory, matrix (A − A T ) is called skew symmetric adjacency matrix of an oriented graph [7] and matrix A , where A k = −A k = i if there is an oriented arc from v k to v and A k = A k = 1 if there is an edge linking v k and v , is called the Hermitian-adjacency matrix of mixed graphs (directed graphs with arcs and edges) [14,11]. It natural to consider the generalized adjacency matrix (zA − z * A T ) with z = θ e iα ∈ C for oriented graphs, which is also Hermitian.…”