On $\alpha$-spectral theory of a directed k-uniform 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.

Keywords

• directed hypergraph
• adjacency tensor
• laplacian tensor
• signless laplacian tensor
• eigenvalue
• $\alpha$-spectral theory
• odd-bipartite hypergraph