Algebraic Properties Of Powers Of Edge Ideas Of Vertex-weighted Oriented Unicyclic Graphs

In this thesis,we mainly consider some algebraic properties,such as regularity and depth,of the powers of edge ideals of vertex-weighted oriented unicyclic graphs,our results extend some previous ones.The main contents are summarized as follows:Let t be a positive integer and D=(V(D),E(D),w)be a vertex-weighted oriented unicyclic graph with some non-trivial weights(i.e.,the weight of vertices of D is not always equal to 1),where V(D)={x1,X2,…,xn}.Set S=K[x1,x2,…,xn]is a polynomial ring in n variables over a field K with x1,x2,…,xn as variables,I(D)(?)S is the edge ideal of D and w=max{w(ax)|x ∈ V(D)}.Firstly,assume that D=Cn is a vertex-weighted oriented cycle with n vertices.We provide an upper bound for the regularity of I(Cn)t,which is reg(?)1)(w+1)+[n+1/3].Secondly,assume that D is a vertex-Ieighted oriented unicyclic graph satisfying w(x)≥ 1 if d(x)≠ for any x ∈ V(D),we prove that reg(I(D)t)=(?)w(x)-|E(D)|+1+(t-1)(w+1),that is,reg(I(D)t)is a linear function with t as a variable.Thirdly,we provide some properties of the sets of associated prime ideals of I(D)t as follows:(i)if D=Cn is a vertex-weighted oriented cycle,then Ass(I(Cn))(?)Ass(I(Cn)t),moreover,Ass(I(Cn))=Ass(I(Cn)t)when w(x)>1 for any x ∈ V(Cn);(ii)if D is a vertex-weighted oriented unicyclic graph with some leaves,then 1(D)has persistence property,that is,Ass(I(D)t)(?)Ass(I(D)t+1).Finally,using above properties,we obtain a lower bound for the depth of I(D)t,which is depth(I(D)t)≥ |V(D)|-|E(D)|+1,and,we prove that depth(I(D)t)gets this lower bound when w(x)>1 if d(x)≠1 for any x ∈ V(D).Moreover,when D=Cn is a vertex-weighted oriented cycle,we prove that if p=(p1,p2,…,pk)is the weight sequence of Cn,then depth(I(Cn)t)≤ n-(?)(2 ×[ni/3])-s-s’+1,where ni=pi+i-pi for any 1≤i ≤k-1,nk=n+p1-pk,s=|{ni:[ni/3]-1=0,1 ≤i≤k}|and s’=|{ni:ni=2 mod 3,1 ≤i ≤k}|.Besides,we get upper and lower bounds for the projective dimension of I(D)t by Auslander-Buchsbaum formula.