A new graphic polynomial, μ (G, t), has been introduced which depends on a vectort and which reduces to the characteristic and matching polynomial of the graphG for certain choices oft. This leads to a unification of a great part of the previously developed theories of the characteristic and matching polynomials. The basic properties of μ (G, t) are determined. A method has been developed by which a functionJ (t) can be associated with every π-electron indexJ. Since the componentt a of the vectort is interpreted as the extent by which a particular cycleZ a of the molecular graphG influences the polynomial μ(G, t), it is possible to use the functionJ(t) in the study of the effect ofZ a on the π-electron indexJ. Total π-electron energy, π-electron charge and π-electron bond order have been analysed by this method and a number of topological rules of the modulo 4 type have been formulated. It is indicated that all these rules have a common algebraic background.