Computing all Laplacian H-eigenvalues for a k-uniform loose path of length three

TL;DR

This paper computes all Laplacian H-eigenvalues for k-uniform loose paths of length three, revealing spectral properties and convergence behavior as the uniformity parameter increases.

Contribution

It provides the first complete characterization of Laplacian H-eigenvalues for this class of hypergraphs, including explicit counts and convergence analysis.

Findings

Number of eigenvalues is 7 for odd-uniform and 14 for even-uniform loose paths.

Eigenvalues converge to {0,1,1.5,2} as k approaches infinity.

Numerical results demonstrate the efficiency of the proposed computation method.

Abstract

The spectral theory of Laplacian tensor is an important tool for revealing some important properties of a hypergraph. It is meaningful to compute all Laplacian H-eigenvalues for some special $k$-uniform hypergraphs. For an odd-uniform loose path of length three, the Laplacian H-spectrum has been studied. However, all Laplacian H-eigenvalues of the class of loose paths have not been found out. In this paper, we compute all Laplacian H-eigenvalues for the class of loose paths. We show that the number of Laplacian H-eigenvalues of an odd(even)-uniform loose path with length three is $7$($14$). Some numerical results are given to show the efficiency of our method. Especially, the numerical results show that its Laplacian H-spectrum converges to $\{0,1,1.5,2\}$ when $k$ goes to infinity. Finally, we establish convergence analysis for a part of the conclusion and also present a conjecture.