On the spectral radius of uniform weighted hypergraph

TL;DR

This paper investigates the spectral radii of adjacency, Laplacian, and signless Laplacian tensors of connected uniform hypergraphs, providing bounds, properties, and relationships among their eigenvalues.

Contribution

It introduces new bounds and properties for the eigenvalues of hypergraph tensors, enhancing understanding of their spectral characteristics.

Findings

Established bounds for eigenvalues of hypergraph tensors.

Characterized extremal eigenvalues for Laplacian and signless Laplacian tensors.

Revealed relationships among eigenvalues of different hypergraph tensors.

Abstract

Let $\mathbb{Q}_{k,n}$ be the set of the connected $k$-uniform weighted hypergraphs with $n$ vertices, where $k,n\geq 3$. 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.