Consider a linear equation system $$ \sum_{j \in J} A_{i,j}.x_{j}=b_{i},\ \ \ \ \forall i\in I. $$
Consider the graph $G(I \cup J,S)$, where $S=\lbrace s:A_{i,j} \ne 0\rbrace $.
What type of information about the solutions to the system can be learned from this graph ?
What properties of this graph do the usual factorization or elimination procedures change ?