On matrix and lattice ideals of digraphs

‎Let G be a simple‎, ‎oriented connected graph with n vertices and m edges‎. ‎Let I(B) be the binomial ideal associated to the incidence matrix \textbf{B} of the graph G‎. ‎Assume that IL is the lattice ideal associated to the rows of the matrix B‎. ‎Also let Bi be a submatrix of B after removing the i-th row‎. ‎We introduce a graph theoretical criterion for G which is a sufficient and necessary condition for I(B)=I(Bi) and I(Bi)=IL‎. ‎After that we introduce another graph theoretical criterion for G which is a sufficient and necessary condition for I(B)=IL‎. ‎It is shown that the heights of I(B) and I(Bi) are equal to n−1 and the dimensions of I(B) and I(Bi) are equal to m−n+1; then I(Bi) is a complete intersection ideal‎.