On the N-spectrum of oriented graphs

Abstract

<jats:title>Abstract</jats:title> <jats:p>Given any digraph <jats:italic>D</jats:italic>, its non-negative spectrum (or N-spectrum, shortly) consists of the eigenvalues of the matrix <jats:italic>AA</jats:italic> <jats:sup> <jats:italic>T</jats:italic> </jats:sup>, where <jats:italic>A</jats:italic> is the adjacency matrix of <jats:italic>D</jats:italic>. In this study, we relate the classical spectrum of undirected graphs to the N-spectrum of their oriented counterparts, permitting us to derive spectral bounds. Moreover, we study the spectral effects caused by certain modifications of a given digraph.</jats:p>

On the Cospectrality of Hermitian Adjacency Matrices of Mixed Graphs

“…Recently, many researchers have proposed other Hermitian adjacency matrices of mixed graphs. For example in [2] and in [1] authors studied the singular values of the traditional adjacency matrix of a mixed graph D, in these papers it was very clear that the singular values of a mixed graph are related to the common out neighbors between vertices. Coincidentally, Guo and Mohar in [3] defined an interesting Hermitian adjacency matrix of mixed graphs as follows: For a mixed graph D, the i−Hermitian adjacency matrix of…”

Hermitian Adjacency Matrices of Mixed Graphs

“…Quite recently, the idea has been presented to modify the definition of the adjacency matrix of a directed graph, some authors studied the matrix AA * , see [1,2], and others used complex numbers, in such a way that it still properly reflects the adjacency relation but at the same time constitutes a Hermitian matrix. Let us give an overview of some efforts and results in this direction.…”