Tutte Polynomials Of A Class Of Signed Graphs And Resonance Cluster Of Planar Arrangements

In this paper,we study Tutte polynomials of two kinds of wheel graph.The twisted wheel graph can be constructed from the complete graph on 4 vertices,K4.Define V(K4)={a,b,c,d}.Proceed by subdividing the edges ac and bd by the addition of one or more vertices.Then each new vertex on edge ac is joined to vertex b.Similarly,each new vertex on each bd is joined to vertex c.The double half-wheels is a graph obtained from the graph G with vertex set {v1,v2,v3,v4,v5} by subdividing the edges v1v5 and v4v5,and then joining each of the new vertices on v1v5 and v4v5 to v2 and v3,respectively.These wheel graphs all have an strap edge and are composed of triangles.As the number of triangles increases,the calculation of Tutte polynomials becomes more and more complex.In order to study the Tutte polynomials of this class of graph,the properties and characteristics of their coefficients are analyzed.We decompose the Tutte polynomials of this class of graph into Tutte polynomials of some base graphs,and transform the study of Tutte polynomials of this class of graph into the study of Tutte polynomials of base graphs.In this paper,Tutte polynomials of twisted wheels with a negative edge and double half-wheels with a negative edge are studied respectively.The Tutte polynomials of signed twisted wheels and signed double half-wheels are calculated by using the deletion-restriction theorem of signed graphs.We use the transformation function in the deletion-restriction theorem of signed graphs,Several regular basic graphs(called base graphs)have been found.Recursive formulas of Tutte polynomials for these basic graphs are derived.Then the Tutte polynomials of this class of signed graphs are obtained with the aid of computer.Furthermore,the characteristic polynomials of the class of signed graphs are obtained.Finally,the OS algebraic dimension of this class of signed graphic arrangements is obtained.As another important algebraic property of hyperplane arragement,which is resonance clusters,have also been extensively studied by mathematicians.Finally,we study the first cohomology and its resonance clusters of a class of hyperplane arragement.We obtain that the first resonance cluster of the B3 hyperplane arrangement is 0.And the resonance cluster dimension of a class of regular hyperplane arrangement in three-dimensional space have a certain regularity.