University of IsfahanTransactions on Combinatorics2251-86577220180601On matrix and lattice ideals of digraphs35462232010.22108/toc.2017.105701.1510ENHamidDamadiDepartment of Mathematics, Amirkabir University of Technology (Tehran Polytechnic) Tehran, Iran.FarhadRahmatiAmirkabir University of TechnologyJournal Article20170802Let $\textit{G}$ 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 $n-1$ and the dimensions of $I(\textbf{B})$ and $I(\textbf{B}_i)$ are equal to $m-n+1$; then $I(\textbf{B}_i)$ is a complete intersection ideal.https://toc.ui.ac.ir/article_22320_b7155094bae6e4bfec0b32c67a2295ec.pdf