Arithmetical Rank Of The Edge Ideals Of Some Graphs

In this paper,we mainly compute the arithmetical rank of the edge ideals of n-cyclic graph with n-1 common vertexes,the main result as following:First,we compute the upperbound of the arithmetical rank of the edge ideals of n-cyclic graph with n-1 common vertexes:Theorem 2.1.1 In n-cyclic graph with n-1 common vertexes,n cycles respectively are C1,...,CN,where V(Ci)={x1i,...,xnii} and Ci intersect Ci+1 at one vertex,if V(CN)≡ 1(mod3),reserve the order ofC1,...,CN,otherwise,keep the original order.For i = 1,...,[N/2],if there is one cycle whose the number of vertexes ≡ 1(mod 3)in C2i-1,C2i,let di = 1,then the edge ideal I(G)satisfy ara(I(G))≤2|V(G)| + 2N-2-M2+M/3-d.Where Mj is the number of which the number of vertexes ≡ j(mod 3)in all cycles,and d=(?).Then,we compute the arithmetical rank of the edge ideals of 3-cyclic graph with 2 common vertexes in two cases:two vertexes of intersection are adjoin or not,ve get following results:Corollary 2.2.3 For 3-cyclic graph with 2 common vertexes G,if two vertexes of intersection are adjoin,let three cycles respectively are Cn,Ck+l,Cm,then(1)If n = 1(mod 3),m = 1(mod 3),when k ≡ 0(mod3),ara(I(G))-bight(I(G))≤ 3,otherwise ara(I(G))-bight(I(G))≤ 2.(2)if k ≡ 0(mod3),m ≡ 1(mod3),when n ≡ 0(mod3)or n = 2(mod3),ara(I(G))-bight(I(G))≤ 2.(3)otherwise,ara(I(G))-bight(I(G))≤ 1.Corollary 2.2.5 For 3-cyclic graph with 2 common vertexes G,if two vertexes of intersection are not adjoin,suppose these two vertexes divided the middle cycle into two parts,respectively are {x1,p2,...,z1},{x1,q2,...,q1,z1} then in n,m,k,l,if there are more than two of them mod 3 remain 0,bight(I(G))= pd(G)= ara(I(G)).if there are more than two of them mod 3 remain 2,ara(I(G))-bight(I(G))≤2.otherwise,ara(I(G))-bight(I(G))≤ 1.Finally,we use induction to promote Theorem 2.2.4 to n-cyclic graph with n-1 common vertexes:Corollary 2.3.2 For n-cyclic graph with n-1 common vertexes G,if for any two vertexes of intersection are not adjoin,suppose these vertexes divided the middle part of the graph into 2n-4 parts,each part without endpoints has k1-1,...,k2N-4-1 vertexes respectively,the first cycle is Cn and the last cycle is Cm,then when all of k1,...,k2N-4 mod 3 remain 0,pd(G)= ara(I(G))=2|V(G)|+2N-2/3,when all of k1,...,k2N-4 mod 3 remain 1,ara(I(G))-pd(G)≤N-[N/2],when all of k1,...,k2N-4 mod 3 remain 2,ara(I(G))-pd(G)≤ N-2-[N/2].Similarly,we have obtained the relevant conclusions of a class of n-cyclic graph with 2 common vertexes:Corollary 2.3.4 For n-cyclic graph with 2 common vertexes G,if N1 cycles are intersect at x1,where the number of vertexes of each cycle is n1,...,nN1,N2 cycles are intersect at z1,where the number of vertexes of each cycle is m1,...,mN2,x1 and z1 are connected by two lines Lk|1 and Ll|1,then(1)when all the number of vertexes mod 3 remain 0,pd(G)= ara(I(G))=2|V(G)|= + 2N1-2N2/3.(2)when all the number of vertexes mod 3 remain 1,pd(G)-bight(I(G))≤N1+ N2＋2.(3)when all the number of vertexes mod 3 remain 2,pd(G)= ara(I(G))=2|V(G)|+N1+ N2-1/3.