Text[" Incidence matrix Aa[i,j] -> graph G(i,j); i nodes, j edges
Aa[i,j]={ 1, if edge j is directed towards node i,
Aa[i,j]={-1, if edge j is directed from node i,
Aa[i,j]={ 0, if edge j is not incident to node i. "]
Aa = {
{-1, -1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1},
{ 0, 1, 0, 0, 0, -1, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0},
{ 1, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0},
{ 0, 0, 1, 1, -1, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0},
{ 0, 0, 0, 0, 1, 0, -1, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0},
{ 0, 0, 0, 0, 0, 1, 1, 0, -1, -1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0},
{ 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, -1, 0, 0, -1, 0, 0, 0, 0, 0},
{ 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, -1, 0, 0, 0, 0, -1, 0},
{ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, -1, 0, -1, 0, 0},
{ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, -1, 0, 0, 0},
{ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1}};
A = Drop[Aa, {1, 1}] ;
MatrixForm[A]
K = A.Transpose[A];
MatrixForm[K]
{lu, p, c} = LUDecomposition[K];
det = Tr[lu, Times]
V = det*Inverse[K];
MatrixForm[V]
W = Apply[GCD, V, {1}]
Y = MatrixForm[V/W]
F = Transpose[A].V.A;
MatrixForm[F]
R = Apply[GCD, F, {1}]
B = MatrixForm[F/R]