The non-negative spectrum of a digraph

Abstract

<jats:title>Abstract</jats:title> <jats:p>Given the adjacency matrix <jats:italic>A</jats:italic> of a digraph, the eigenvalues of the matrix <jats:italic>AA<jats:sup>T</jats:sup></jats:italic> constitute the so-called non-negative spectrum of this digraph. We investigate the relation between the structure of digraphs and their non-negative spectra and associated eigenvectors. In particular, it turns out that the non-negative spectrum of a digraph can be derived from the traditional (adjacency) spectrum of certain undirected bipartite graphs.</jats:p>

On the N-spectrum of oriented graphs

“…With this image before our eyes, we define the rows = {( )} = … k j , Proof. The claim is implicit in the proof of Corollary 4 in [11]. To prove it, we recall the well-known fact (see e.g.…”

“…When it comes to analyzing the N-spectrum of a digraph, it is possible to conveniently choose a vertex order such that the matrix AA T assumes the block diagonal form. Following [11], we define a zig-zag trail (of length 2l) between two vertices x and y to be a sequence of arcs…”

“…Theorem 1. (Square Theorem [11]). Let D be a bipartite digraph such that each vertex is either a source or a sink.…”

“…It is interesting to study how modifying a given digraph reflects in changes of its N-spectrum. Several modifications have already been considered in [10] and [11]. In this section, we want to focus on N-eigenvalue interlacing.…”

Inverse of $α$-Hermitian Adjacency Matrix of a Unicyclic Bipartite Graph

“…The author called them non negative spectrum of digraphs. In [2], authors proved that the non negative spectrum is totally controlled by a vertex partition called common out neighbor partition. Authors in [3] and in [4] (independently) proposed a new adjacency matrix of mixed graphs as follows: For a mixed graph X, the hermitian adjacency matrix of…”