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 ($k\ge 3$) supertrees with fixed number of vertices, the supertrees with the maximum, the second maximum and the minimum spectral radius are completely determined, respectively.