**Authors:** Gholam-Hasan Shirdel, Ameneh Mortezaee, Effat Golpar-Raboky

**Keywords:** Directed Hypergraph, Adjacency tensor, Laplacian tensor, Signless Laplacian tensor, Eigenvalue, $\alpha$-spectral theory, Odd-bipartite Hypergraph.

### Abstract

In this paper, we study a k-uniform directed hypergraph in general form and introduce its adjacency tensor, Laplacian tensor and signless Laplacian tensor. For the $k$-uniform directed hypergraph $\mathcal{H}$ and $0\leq \alpha <1$ the convex linear combination of $\mathcal{D}$ and $\mathcal{A}$ has been defined as $\mathcal{A}_\alpha=\alpha\mathcal{D}+(1-\alpha)\mathcal{A}$, where $\mathcal{D}$ and $\mathcal{A}$ are the degree tensor and the adjacency tensor of $\mathcal{H}$, respectively. We propose some spectral properties of $\mathcal{A}_\alpha$. We also introduce power directed hypergraph and cored directed hypergraph and investigate their $\alpha$-spectral properties.

Department of Mathematics, University of Qom

Qom, I. R. Iran

E-mail: , ,

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