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

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 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.