Spanning Simplicial Complexes And The Arithmetical Rank Of Edge Ideals Of Some N-Cyclic Graphs With A Common Edge

In our paper, we mainly study the properties of monomial ideals by the tool of graph theory. We divide our paper into two parts:Firstly, we mainly study some algebraic properties of spanning simplicial complexes of the n-cyclic graphs Gt1, t2, ···, tnwith a common edge, under the condition that the length of every cyclic graph Gtiis t, we give a formula for f-vector of ?s(Gt1, t2, ···, tn); Moreover we give a formula for Hilbert series of the Stanley-Reisner ring K[?s(Gt1, t2, ···, tn)] of?s(Gt1, t2, ···, tn), where K is a field.The main results of this part are following:Theorem Let ?s(Gt1, t2,..., tn) be the spanning simplicial complex of the n-cyclic graphs Gt1, t2, ···, tnwith a common edge, we set the length of every cyclic graph Gtiis t where i ∈ {1,..., n}, and a number of the edges of graphs Gt1, t2, ···, tnis b = n(t- 1) + 1,then the dimension and the f-vector of ?s(Gt1, t2,..., tn) are given by d = n(t- 2) and f =(f0, f1,..., fd), where where m = [j+1t], and we adopt the following convention:0i=1gi= 0,0i=2hi= 0,1i=2hi= 0.Secondly, for the sake of simplicity, we can denote by G the n-cyclic graphs Gt1, t2, ···, tnwith a common edge. We give some upper bounds on ara(I(G)) and some lower bounds on bight(I(G)) of edge ideals I(G) of some graphs G, moreover, by the inequalities bight(I(G)) ≤ pdR(R/I(G)) ≤ ara(I(G)), we can discuss when pdR(R/I(G)) =ara(I(G)) is true.The main results of this part are following:Theorem 1 Let G be a k1-cyclic graph consisting of the union of k1 cycles G3r1,...,G3rk1with a common edge x1x2, then ara(I(G)) ≤k1i=1(2ri- 1) + 1, bight(I(G)) ≥k1i=1(2ri- 1) + 1, and moreover bight(I(G)) = pdR(R/I(G)) = ara(I(G)).Theorem 2 Let G be a k2-cyclic graph consisting of the union of k2 cycles G3s1+1,...,G3sk2+1with a common edge x1x2,(1) if si= 1, ? i ∈ {1, 2,..., k2- 1}, then ara(I(G)) ≤ k2+ 2sk2, bight(I(G)) =k2+ 2sk2, and moreover bight(I(G)) = pdR(R/I(G)) = ara(I(G)).(2) if G has at most k2- 2 G4 s, then ara(I(G)) ≤k2i=1(2si) + 1, bight(I(G)) =k2i=1(2si)- k2+ 2, and moreover ara(I(G))- bight(I(G)) ≤ k2- 1.Theorem 3 Let G be a k3-cyclic graph consisting of the union of k3 cycles G3t1+2,...,G3tk3+2with a common edge x1x2, then ara(I(G)) ≤k3i=1(2ti) + 1, bight(I(G)) ≥k3i=1(2ti) +1, and moreover bight(I(G)) = pdR(R/I(G)) = ara(I(G)).Theorem 4 Let G be an n-cyclic graph consisting of the union of n cycles G3r1,...,G3rk1, G3s1+1,..., G3sk2+1, G3t1+2,..., G3tk3+2with a common edge x1x2, where k1+ k2+k3= n, then ara(I(G)) ≤k1i=1(2ri- 1) +k2i=1(2si) +k3i=1(2ti) + 1,where we adopt the following convention: whenever, in a sum, the index runs from 1 to 0,the sum has to be taken equal to zero, and moreover when k2= 0, we have bight(I(G)) =pdR(R/I(G)) = ara(I(G)).