The principal eigenvector to $\alpha$-spectral radius of hypergraphs

Abstract

For a connected hypergraph H with $rank(H)=r$ , let $\mathcal {D}(H)$ and $\mathcal {A}(H)$ be the diagonal tensor of degrees and the adjacency tensor of H, respectively. For $0 \le \alpha < 1$, the $\alpha$-spectral radius $\rho _{\alpha }(H)$ of H is defined as $\rho _{\alpha }(H)=\max \{x^{T}(\mathcal {A}_{\alpha }x)|x \in \mathbf {R}_{+}^{n},\Vert x\Vert _{r}=1\}$, where $\mathcal {A}_{\alpha }(H)=\alpha \mathcal {D}(H)+(1-\alpha )\mathcal {A}(H)$. In this paper, we present some bounds on entries of the positive unit eigenvector corresponding to the $\alpha$-spectral radius of connected uniform hypergraphs. Furthermore, we obtain some bounds on entries of the positive unit eigenvector corresponding to the $\alpha$-spectral radius of connected general hypergraphs.