The Laplacian Spectral Moments Of Power Hypergraphs And Their Application

Spectral hypergraph is an important research direction in graph theory and combinatorial mathematics.The spectral moment of hypergraph is a powerful tool to study the spectrum of hypergraph,and the structure of hypergraph can be described by its spectral moment.The d-th order Laplacian spectral moment of a uniform hypergraph is equal to the sum of d-th powers of all eigenvalues of its Laplacian tensor.Since the d-th order trace of a tensor is equal to the sum of the d-th powers of all eigenvalues,the d-th order Laplacian spectral moment of a hypergraph is equal to the d-th order trace of its Laplacian tensor.Therefore,the Laplacian spectral moment of a hypergraph can be studied by the trace of tensor.The k-th power hypergraph G^kof G is the k-uniform hypergraph obtained by adding k-2 new vertices whose degree are 1 to each edge of G.In this thesis,Laplacian moments and signless Laplacian moments of power hypergraphs are studied.The expressions of the 2k-th order Laplacian moment and signless Laplacian moment of the k-power hypergraphs are given,and these expressions are represented by the degree sequence of graphs.And the graph parameter expressions of the 5-th order Laplacian moment and signless Laplacian moment of the 3-power hypergraph are given.According to the graph parameter expressions of Laplacian moments and signless Laplacian moments of power hypergraphs,the same graph parameters of power hypergraphs with the same Laplacian spectrum or signless Laplacian spectrum are given.Two hypergraphs are（signless）Laplacian cospectral if and only if any order trace of their（signless）Laplacian tensor is equal,and the tensor spectrum of power hypergraph contains more graph structure information than the matrix spectrum of graph.In this thesis,we use（signless）Laplacian spectrum of power hypergraphs to study the spectral characterization of some graphs.It is proved that rose graphs and C₆×K_r are determined by the Laplacian spectrum of their 3-power hypergraphs;∞-graphs,K₃ ∪ rK₁ and K_1,3×H are determined by the signless Laplacian spectrum of their 3-power hypergraphs;the corona C_n（?）2K₁ are determined by the signless Laplacian spectrum of their 4-power hypergraphs.