Research On The Representation Of Characteristic Polynomials Of Some Hypertrees

Spectral theory of hypergraphs is an important research field in graph theory and combinatorial mathematics,which has been widely applied in information science,life science,complex network analysis,artificial intelligence and other fields.The spectral hypergraph theory mainly researches the relationship between the adjacent tensor,Laplacian tensor,signless Laplacian tensor eigenvalue,eigenvector and hypergraph structure.The spectrum of a hypergraph is the spectrum of its adjacency tensor,and the characteristic polynomial of a hypergraph is the characteristic polynomial of its adjacency tensor.In recent years,the research on characteristic polynomials of hypergraphs has attracted the attention of many scholars.In this paper,we mainly research the characteristic polynomials of adjacency tensor of some hypertrees.The resultant Poisson product formula is used to study the adjacency tensor of the hypergraph,and combined with the structure of the hypergraph,the expressions of the characteristic polynomials of the starlike hypertrees and the recurrences of the characteristic polynomials of the three kinds of double quasi-star hypertrees are given.Starlike hypertree is a hypertree with only one point whose degree is greater than 2.In this paper,we use the recurrence formulas of the characteristic polynomials of starlike tree and starlike hypertree,the recurrence formula of the characteristic polynomial of starlike hypertree is expressed by the eigenvalue of starlike tree,so that the characteristic polynomial of starlike hypertree is expressed in the form of factor product,and the expression of the characteristic polynomial of k-uniform starlike hypertree is obtained,whose degree of center point is m.Double quasi-star hypertrees is a hypertree with only two points whose degree is greater than 2.The characteristic polynomial of hypertrees is the characteristic polynomial of its adjacency tensor,which is the resultant of polynomial system corresponding to its adjacency tensor.In this paper,by using the resultant Poisson product formula,the expressions of three special characteristic polynomial recursions of 3-uniform double quasi-star hypertrees are given.