# Non-negative spectrum of a digraph

### Irena Jovanovi\' c School of Computing, Union University

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Minisymposium: SPECTRAL GRAPH THEORY

Content: In this talk, digraphs will be considered by means of eigenvalues of the matrix $AA^T$ (or $A^TA$) where $A$ is the adjacency matrix of a digraph. The common spectrum of these matrices is called the \emph{non-negative spectrum} or the \emph{$N$-spectrum} of a digraph. Several properties of the $N$-spectrum together with a family of $N$-cospectral digraphs will be presented. The notion of cospectrality w.r.t. a graph matrix will be generalized to the cospectrality w.r.t. several graph matrices. In accordance with this, a family of cospectral multigraphs with respect to the adjacency matrix will be described. These multigraphs are isomorphic when they are associated to a circulant digraph. Some examples of the $(N,Q)$-cospectral mates of digraphs and multigraphs, where $Q$ is the signless Laplacian matrix, will be exposed.

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