The Spectra Of The(Signless)normalized Laplacian Of Power Hypergraphs

Let H=(VH,EH)be a general graph,for any r≥3,the r-th power of H is called r-th power hypergraph,which can be represented as Hr=(VHr,EHr).A power hypergraph can be regarded as an r-uniform hypergraph with exactly r vertices on each edge of an ordinary graph after adding r-2 vertices on each edge.With the help of the relationship between the eigenvalues of power hypergraphs and the eigenvalues of original graphs,the problems on more complex hypergraphs can be studied by using the parameters of graphs.In 2015,Hu and Qi gave the definition of the normalized Laplacian tensor of hypergraphs,and discussed some spectral properties of the normalized Laplacian tensor.Let Hr be an r-uniform hypergraph has n vertices.The normalized adjacency tencer AHr=(ai1i2…ir),which is an r-th order n-dimension tensor of Hr,is defined as#12 When Hr has no isolated vertex,the normalized Laplacian tensor of hypergraph Hr is defined as LHr=L-AHr,where L be the unit tensor.In 2017,Shao et al gave the expression of the signless normalized Laplacian tensor.This paper gives relevant conclusions based on the above definitionsIn this paper,the vertices from the original graph are called main vertices and the added vertices after generating the power hypergraph are called additional vertices.When the eigenvector component corresponding to the additional vertices is not zero,conditions are given for the transformation between the eigenvalues of the(signless)normalized Laplacian of power hypergraph and the(signless)normalized Laplacian of the original graph.The relationship between the components of eigenvectors is given.The theorem is applied to some special graphs and some examples are given.