IncidenceMatrix(G) returns the incidence matrix of a graph G whose rows are indexed by the vertices and columns by the edges of G.  The order of the edges is defined by Edges(G).

If G is undirected the entry $i,j$ of this $\left\{0\,1\right\}$-matrix is $1$ iff vertex i is incident to edge j.

If G is directed the entry $i,j$ of this $\left\{\mathrm{-1}\,0\,1\right\}$-matrix is $1$ iff vertex i is the head of arc j and $\mathrm{-1}$ iff vertex i is the tail of arc j.

If G is directed and reverse is specified, the entry $i,j$ of this $\left\{\mathrm{-1}\,0\,1\right\}$-matrix is $\mathrm{-1}$ iff vertex i is the head of arc j and $1$ iff vertex i is the tail of arc j.

$\mathrm{with}\left(\mathrm{GraphTheory}\right)\:$

$G≔\mathrm{CompleteGraph}\left(4\right)$

$G≔\mathrm{Graph\; 1:\; an\; undirected\; unweighted\; graph\; with\; 4\; vertices\; and\; 6\; edge(s)}$

$\mathrm{Edges}\left(G\right)$

$\left\{\left\{1\,2\right\}\,\left\{1\,3\right\}\,\left\{1\,4\right\}\,\left\{2\,3\right\}\,\left\{2\,4\right\}\,\left\{3\,4\right\}\right\}$

$\mathrm{IncidenceMatrix}\left(G\right)$

$\left[\begin{array}{cccccc}1& 1& 1& 0& 0& 0\\ 1& 0& 0& 1& 1& 0\\ 0& 1& 0& 1& 0& 1\\ 0& 0& 1& 0& 1& 1\end{array}\right]$

$\mathrm{DG}≔\mathrm{Digraph}\left(\mathrm{Trail}\left(1\,2\,3\,4\,5\,3\right)\,\mathrm{Trail}\left(1\,5\,2\,4\,1\right)\right)$

$\mathrm{DG}≔\mathrm{Graph\; 2:\; a\; directed\; unweighted\; graph\; with\; 5\; vertices\; and\; 9\; arc(s)}$

$\mathrm{Edges}\left(\mathrm{DG}\right)$

$\left\{\left[1\,2\right]\,\left[1\,5\right]\,\left[2\,3\right]\,\left[2\,4\right]\,\left[3\,4\right]\,\left[4\,1\right]\,\left[4\,5\right]\,\left[5\,2\right]\,\left[5\,3\right]\right\}$

$\mathrm{IncidenceMatrix}\left(\mathrm{DG}\right)$

$\left[\begin{array}{ccccccccc}\mathrm{-1}& \mathrm{-1}& 0& 0& 0& 1& 0& 0& 0\\ 1& 0& \mathrm{-1}& \mathrm{-1}& 0& 0& 0& 1& 0\\ 0& 0& 1& 0& \mathrm{-1}& 0& 0& 0& 1\\ 0& 0& 0& 1& 1& \mathrm{-1}& \mathrm{-1}& 0& 0\\ 0& 1& 0& 0& 0& 0& 1& \mathrm{-1}& \mathrm{-1}\end{array}\right]$

$\mathrm{IncidenceMatrix}\left(\mathrm{DG}\,\mathrm{reverse}\right)$

$\left[\begin{array}{ccccccccc}1& 1& 0& 0& 0& \mathrm{-1}& 0& 0& 0\\ \mathrm{-1}& 0& 1& 1& 0& 0& 0& \mathrm{-1}& 0\\ 0& 0& \mathrm{-1}& 0& 1& 0& 0& 0& \mathrm{-1}\\ 0& 0& 0& \mathrm{-1}& \mathrm{-1}& 1& 1& 0& 0\\ 0& \mathrm{-1}& 0& 0& 0& 0& \mathrm{-1}& 1& 1\end{array}\right]$