A note on the circuit polynomials and characteristic polynomials of wheels and ladders

Let G be a graph. The circuit polynomial of G is the polynomial Σ Π_αw_α, where w_α is a weight given to the circuit α in G, Π_αw_α is the weight of a spanning subgraph of G, consisting of circuits, nodes and edges, and the summation is taken over all such spanning subgraphs in G. The circuit polynomial of a graph is a generalization of its characteristic polynomial. The characteristic polynomials of wheels and ladders are deduced from their corresponding circuit polynomials.