  As an instance of the B-polynomial, the circuit, or cycle, polynomial
P(G(Gamma); w) of the generalized rooted product G(Gamma) of graphs was studied
by Farrell and Rosenfeld ({\em Jour. Math. Sci. (India)}, 2000, \textbf{11}(1),
35--47) and Rosenfeld and Diudea ({\em Internet Electron. J. Mol. Des.}, 2002,
\textbf{1}(3), 142--156). In both cases, the rooted product G(Gamma) was
considered without any restrictions on graphs G and Gamma. Herein, we present a
new general result and its corollaries concerning the case when the core graph
G is restricted to be bipartite. The last instance of G(Gamma), as well as all
its predecessors, can find chemical applications.
