Oriented incidence matrix
= Oriented incidence matrix
{title2=$D_{e,i}=1,\quad D_{e,j}=-1$}
An oriented incidence matrix records an arbitrarily chosen orientation of each <edge> of a <graph>. Use one row per <edge>, with $+1$ and $-1$ at its endpoints and zeros elsewhere; the transpose convention is also common. The product $D\theta$ records endpoint differences. For a <connected graph>, $\ker D=\operatorname{span}\{\mathbf1\}$, since zero difference propagates along every <graph path>.