%0 Journal Article
%T On matrix and lattice ideals of digraphs
%J Transactions on Combinatorics
%I University of Isfahan
%Z 2251-8657
%A Damadi, Hamid
%A Rahmati, Farhad
%D 2018
%\ 06/01/2018
%V 7
%N 2
%P 35-46
%! On matrix and lattice ideals of digraphs
%K Directed graph
%K Binomial ideal
%K Matrix ideals
%R 10.22108/toc.2017.105701.1510
%X 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 $n-1$ 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.
%U https://toc.ui.ac.ir/article_22320_b7155094bae6e4bfec0b32c67a2295ec.pdf