This code produces directed reflexive graphs.
Written by Eric Johnson at FOCUS | numeric.

n=15;
Clear A;
Clear G;
MatA=Table[Random[Integer,{0,RandomInteger[]}],{i,1,n},{j,1,n}];
MatD=MatrixForm[UpperTriangularize[MatA]];
AD=DiagonalMatrix[Diagonal[MatA]];
A = MatA;
G=AdjacencyGraph[A,VertexLabels->"Name",ImagePadding->10,ImageSize->550]
Table[AdjacencyGraph[A,GraphLayout->l,PlotLabel->l,ImageSize->350,VertexLabels->"Name"],{l,{"CircularEmbedding","SpiralEmbedding"}}]
MatrixForm[A]
EdgeCount[G]
EdgeList[G]
VertexCount[G]
VertexList[G]
MatrixPlot[A,ImageSize->350]
1.*Eigenvalues[A]
1.*Tr[A]
1.*Det[A]
PA[t_]=CharacteristicPolynomial[A,t]
Plot[PA[t],{t,-5,5},PlotRange->{{-10,10},{-25,25}},ImageSize->550]
ScientificForm[PA[1.Eigenvalues[A]]]