On matrix and lattice ideals of digraphs

Abstract

‎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.