Matrix analysis of oriented graphs with irreducible feedback loops

THE PROBLEM of determining the possible I closed circuits available in a situation in which n terminals are interconnected in some fashion by unilateral branches is one of considerable importance in a number of diverse fields. It may be regarded as that of a topological characterization of a linear network, with specific branch gains between terminals; in such a case, the system may be designated (see Fig. 1) by the branch gain or transmission factor matrix A = (aij), in terms of which the terminal inputs Ei and terminal “voltages” Vi are related by