Hamid Damadi - Department of Mathematics, Amirkabir University of Technology (Tehran Polytechnic) Tehran, Iran.
Farhad Rahmati - Amirkabir University of Technology

‎Let \textit{G} be a simple‎, ‎oriented connected graph with n vertices and m edges‎. ‎Let I(\textbf{B}) be the binomial ideal associated to the incidence matrix \textbf{B} of the graph G‎. ‎Assume that I_L is the lattice ideal associated to the rows of the matrix \textbf{B}‎. ‎Also let \textbf{B}_i be a submatrix of \textbf{B} after removing the i-th row‎. ‎We introduce a graph theoretical criterion for G which is a sufficient and necessary condition for I(\textbf{B})=I(\textbf{B}_i) and I(\textbf{B}_i)=I_L‎. ‎After that we introduce another graph theoretical criterion for G which is a sufficient and necessary condition for I(\textbf{B})=I_L‎. ‎It is shown that the heights of I(\textbf{B}) and I(\textbf{B}_i) are equal to n-۱ and the dimensions of I(\textbf{B}) and I(\textbf{B}_i) are equal to m-n+۱; then I(\textbf{B}_i) is a complete intersection ideal‎.

Directed graph, Binomial ideal, Matrix ideals