The cycle (circuit) polynomial of a graph with double and triple weights of edges and cycles

Farrell introduced the general class of graph polynomials which he called the family polynomials, or F-polynomials, of graphs. One of these is the cycle, or circuit, polynomial. This polynomial is in turn a common generalization of the characteristic, permanental, and matching polynomials of a graph, as well as a wide variety of statistical-mechanical partition functions, such as were earlier known. Herein, we specially derive weighted generalizations of the characteristic and permanental polynomials requiring for calculation thereof to assign double (res. triple) weights to all Sachs subgraphs of a graph. To elaborate an analytical method of calculation, we extend our earlier differential-operator approach which is now employing operator matrices derived from the adjacency matrix. Some theorematic results are obtained.