The circuit polynomial of the restricted rooted product G(Γ)G(Γ) of graphs with a bipartite core G

As an instance of the B  -polynomial, the circuit, or cycle, polynomial P(G(Γ);w)P(G(Γ);w) of the generalized rooted product G(Γ)G(Γ) of graphs was studied by Farrell and Rosenfeld [Block and articulation node polynomials of the generalized rooted product of graphs, J. of Math. Sci. (India) 11(1) (2000) 35–47] and Rosenfeld and Diudea [The block polynomials and block spectra of dendrimers, Internet Electron. J. Mol. Design 1(3) (2002) 142–156]. In both cases, the rooted product G(Γ)G(Γ) was considered without any restrictions on graphs G   and ΓΓ. Herein, we present a new general result and its corollaries concerning the case when the core graph G is restricted to be bipartite.