Abstract: This paper presents a unified theory of augmented graph. Two basic theorems for a matrix are proved. By making use of graph augmentation, evaluation of the determinant and cofactors associated with a matrix is carried out in one single analysis which is referred to the J-operation, so that is of a set of linear algebraic equations. The augmented Coates graph and Mason graph are topologically simplified. Since the need for the formulas called D-operation is eliminated, the augmented graph serves as a theoretic basis, on which the existing network topology could be reduced greatly.