Extremal spectral radii of uniform supertrees

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, we represent some results on the spectral radius changing under some graphic structural perturbations. With these results, 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.