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 a k-uniform loose path of length three, the Laplacian H-spectrum has been studied when k is odd. However, all Laplacian H-eigenvalues of a k-uniform loose path of length three have not been found out. In this paper, we compute all Laplacian H-eigenvalues for a k-uniform loose path of length three. 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 show that the convergence of Laplacian H-spectrum from theoretical analysis.

中文翻译：

---

计算长度为3的均匀松散路径的所有Laplacian H特征值

拉普拉斯张量的谱理论是揭示超图某些重要性质的重要工具。计算某些特殊的k一致超图的所有Laplacian H特征值是有意义的。对于长度为3的k均匀松散路径，研究了当k为奇数时的拉普拉斯H谱。但是，尚未找到长度为3的k均匀松散路径的所有Laplacian H特征值。在本文中，我们计算了长度为3的k均匀松散路径的所有Laplacian H特征值。我们显示，长度为3的奇（偶）均匀松散路径的Laplacian H-特征值的数目为7（14）。数值结果表明了该方法的有效性。特别是，数值结果表明，其拉普拉斯算子的H谱收敛于\（\ {0，1，1.5，2 \} \）当k达到无穷大时。最后，我们从理论分析证明了拉普拉斯H谱的收敛性。