In this paper, we characterize the structure generating function of a strongly connected automaton. It is proved that poles of the minimal absolute value of the structure generating function are , e2πi/h, , e2πi( h 1)/ h where h is the period of a strongly connected automaton and r is the Frobenius's root of its incidence matrix. This result is a specialization of Berstol's result which characterizes the structure generating function in general. His method is based on the analysis. But we prove our results using some facts in linear algebra and graph theory. And we obtain analogous results for an automaton by considering the strongly connected automaton which contains as a subautomaton.

We give a simple formula which represents the relationship between incident matrices of two transformation semigroups X and Y and the incident matrix of their wreath product X Y. The structure generating function of the wreath product can be easily obtained by using this incident matrix.