Projective Dimension And Regularity Of Powers Of Edge Ideals Of Some Vertex-Weighted Oriented Bipartite Graphs

Let S be a polynomial ring,M be a finitely generated graded S-module,and the projective dimension and regularity of module M can measure the complexity of its minimal graded free resolution.Moreover,according to the formula of Auslander-Buchsbaum,the projective dimension can help us to study the Cohen-Macaulayness of M.Therefore,more and more scholars pay attention to the projective dimension and regularity.Based on the previous work,this paper mainly studies the projective dimension and regularity of powers of edge ideals of two types of vertex-weighted oriented bipartite graphs.The first type of graph is vertex-weighted rooted forest.Let s,t be two positive integers and D=(V(D),E(D),w)be a vertex-weighted rooted forest with s connected components.We consider projection dimension and regularity of powers of edge ideal of D,and obtain their upper bounds respectively,which are pd(I(D)t)?|E(D)|-1 and reg(I(D)t)?(?)(x)-|E(D)|-(s-1)+v(D)+(t-1)(w+1),where v(D)is the induced matching number of D and w=max {w(x)| x ? V(D)}.Furthermore,when D satisfies w(x)>2 if d(x)?1 for any x ? V(D),then(i)pd(I(D)t)equals |E(D)|-1,which is a constant;(ii)reg(I(D)t)equals(?)(x)-|E(D)|+1+(t-1)(w+1),which is a linear function of t.The second type of graph is vertex-weighted oriented gap-free bipartite graph.Let t be a positive integers,D=(V(D),E(D),w)be a vertex-weighted oriented gap-free bipartite graph with bipartition X?{x1,...,xl},Y={y1,...,ym} and the orientation of every edge in D is away from X.We prove that the regularity of powers of edge ideal of D is equal to(?)(x)-|V(D)|?2+(t-1)(w+1),which is xEV(D)a linear function of t,where w=max{w(x)|x?V(D)}.We also provide the upper and lower bounds for the projective dimension of powers of edge ideal of D:max{l-1,m-1} ? pd(I(D)t)?|V(D)|-2.Furthermore,pd(I(D)t)attain the upper if D is a complete bipartite graph and the lower and upper bounds of pd(I(D)t)are the same if D is a star graph.Meanwhile,we also provide some precise formulas for regularity and give the upper and lower bounds for the projection dimension of powers of edge ideals of the disjoint union of some vertex-weighted oriented gap-free bipartite graphs.