On the spectral radius of uniform weighted hypergraph

Abstract

Let ${\mathbb{Q}}_{k,n}$ be the set of the connected $k$-uniform weighted hypergraphs with $n$ vertices, where $k,n\ge 3$. For a hypergraph $G\in {\mathbb{Q}}_{k,n}$, let $\mathcal{A}(G)$, $\mathcal{L}(G)$ and $\mathcal{Q}(G)$ be its adjacency tensor, Laplacian tensor and signless Laplacian tensor, respectively. The spectral radii of $\mathcal{A}(G)$ and $\mathcal{Q}(G)$ are investigated. Some basic properties of the $H$-eigenvalue, the $H^{+}$-eigenvalue and the $H^{++}$-eigenvalue of $\mathcal{A}(G)$, $\mathcal{L}(G)$ and $\mathcal{Q}(G)$ are presented. Several lower and upper bounds of the $H$-eigenvalue, the $H^{+}$-eigenvalue and the $H^{++}$-eigenvalue for $\mathcal{A}(G)$, $\mathcal{L}(G)$ and $\mathcal{Q}(G)$ are established. The largest $H^{+}$-eigenvalue of $\mathcal{L}(G)$ and the smallest $H^{+}$-eigenvalue of $\mathcal{Q}(G)$ are characterized. A relationship among the $H$-eigenvalues of $\mathcal{L}(G)$, $\mathcal{Q}(G)$ and $\mathcal{A}(G)$ is also given.