Damadi, H., Rahmati, F. (2018). On matrix and lattice ideals of digraphs. Transactions on Combinatorics, 7(2), 35-46. doi: 10.22108/toc.2017.105701.1510

Hamid Damadi; Farhad Rahmati. "On matrix and lattice ideals of digraphs". Transactions on Combinatorics, 7, 2, 2018, 35-46. doi: 10.22108/toc.2017.105701.1510

Damadi, H., Rahmati, F. (2018). 'On matrix and lattice ideals of digraphs', Transactions on Combinatorics, 7(2), pp. 35-46. doi: 10.22108/toc.2017.105701.1510

Damadi, H., Rahmati, F. On matrix and lattice ideals of digraphs. Transactions on Combinatorics, 2018; 7(2): 35-46. doi: 10.22108/toc.2017.105701.1510

^{1}Department of Mathematics, Amirkabir University of Technology (Tehran Polytechnic) Tehran, Iran.

^{2}Amirkabir University of Technology

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.