A topological investigation of network determinants

Polynomials associated with network functions are investigated by simple topological methods. The main result is contained in Theorem 1, which states that the determinant of the nodal admittance matrix P of a connected RLC network without transformers is of the form: det P=[Polynomial in ¿ containing a constant term and of degree (2N+2¿SC¿SCR¿SL¿SLR)]/¿N+1¿SL¿SLR. Here, ¿ is the complex frequency variable, N is the number of nodes in the network and SC, SCR, SL and SLR are the connectivities of those sub-networks of the given network formed, respectively, of the capacitors only, the capacitors and resistors only, the inductors only and the inductors and resistors only. This result is based upon an expression for det P as the sum of tree-products, which are defined. A dual result is obtained for the determinant of the loop-impedance matrix, and the extension to other network functions is indicated, the driving-point admittance function being taken as an example.