Orlik-Solomon Algebras And Tutte Polynomials Of A Class Of Graphic Arrangements

We study a special class of graphic arrangements, i.e., hyperplane arrangement associated with an n-rank wheel graph. Suppose C_n is a circle with n vertices. Connecting the n + 1 vertex with every vertices form C_n, we obtain the n-rank wheel graph, i.e.,Α_n, it has n + 1 vertices and 2n edges.Firstly, we prove a formula on the dimension of the Orlik-Solomon algebra of the graphic arrangement associated with the n-rank wheel graph. We use deletion-restriction algorithm to reduce the computation for n-rank wheel graphic arrangement to a chord graphic arrangement. We obtain the following formulae in this paper. LetΑ_n denote the arrangement associated with the n -rank wheel graph. For n≥3 and 0≤p≤n-2, the dimension of the p-th component OS^p(Α_n) of the Orlik-Solomon algebra OS (Α_n) ofΑ_n isHence, the total dimension of the Orlik-Solomon algebra is dim OS(Α_n)=3ⁿ-3。We also study graphs related toΑ_n and show that the dimension of the Orlik-Solomon algebra of the arrangement associated with the new graph is 3ⁿ⁺¹-9.Secondly, we compute Tutte polynomials of the n -rank wheel graphand graphs obtained from n-rank wheel graph by deleting an boundary edge and obtain general formulae. The Tutte polynomials of n -rank wheel graphΑ_n isThirdly, we study two-variable chromatic polynomial for n-rank wheel graph and illustrates that this polynomial and Tutte polynomial are two different concepts.Finally, we consider the computation of cohomologies for arrangements of polygon with n edges.