The Computation Of F-vector Of Spanning Complexes And Arithmetical Ranks Of Its Edge Ideal Of N-Cyclic Graphs With A Common Vertex

In this paper, I mainly ponder on the algebraic properties of spanning complex?s(Gl1,l2,···,ln) and the edge ideal of n-cycle graph Gl1, l2, ···, lnwith a common vertex. Two parts are contained in this paper.In the first part, for a given n-cycle graph Gl1, l2, ···, ln, we gain its spanning trees with cutting-down method. Also we gain the spanning simplicial complex ?s(Gl1, l2, ···, ln) of n-cycle graph Gl1, l2, ···, ln. We also consider some algebraic properties and the formula of its f-vector of ?s(Gl1, l2, ···, ln).In the second part, we consider the arithmetical rank ara(I(G)) of the edge ideals I(G) of n-cyclic graphs G(here G means Gl1, l2, ···, ln) with a common vertex. About the computation of ara(I(G)), we make a classification of the length of the cycle, that is:li≡ 0 mod 3, li≡ 1 mod 3 and li≡ 2 mod 3. And we obtain that when n ≡ 0, 2 mod3, bight(I(G)) = pdR(R/I(G)) = ara(I(G)). Morever, we gain that when n ≡ 1 mod 3,ara(I(G))- bight(I(G)) ≤ k2, where k2 is the number of cycles with length li≡ 1 mod3.