The uniform supertrees with the extremal spectral radius

Abstract: For a $hypergraph$ $\mathcal{G}=(V, E)$ consisting of a nonempty vertex set $V=V(\mathcal{G})$ and an edge set $E=E(\mathcal{G})$, its $adjacency$ $matrix$ $\mathcal {A}{\mathcal{G}}=[(\mathcal {A}{\mathcal{G}}){ij}]$ is defined as $(\mathcal {A}{\mathcal{G}}){ij}=\sum{e\in E_{ij}}\frac{1}{|e| - 1}$, where $E_{ij} = {e \in E : i, j \in e}$. The $spectral$ $radius$ of a hypergraph $\mathcal{G}$, denoted by $\rho(\mathcal {G})$, is the maximum modulus among all eigenvalues of $\mathcal {A}_{\mathcal{G}}$. In this paper, among all $k$-uniform ($k\geq 3$) supertrees with fixed number of vertices, the supertrees with the maximum, the second maximum and the minimum spectral radius are completely determined, respectively.