On a class of polynomials obtained from the circuits in a graph and its application to characteristics polynomials of graphs

Let G be a graph. With each circuit α in G, we can associate a weight w_α, A circuit cover of G is a spanning subgraph of G in which every component is a circuit. With every circuit cover of G, we can associate the monomial Π_αw_α, where the product is taken over all components of the cover. The circuit polynomial of G is ΣΠ_αw_α, where the summation is taken over all circuit covers in G. We show that the characteristics polynomial of G is a special case of its circuit polynomial. Previously obtained and also new results for characteristic polynomials are easily deduced. We also derive the circuit polynomials of various classes of graphs.