Exponents of two-colored digraphs with two cycles

For two nonnegative matrices A and B of the same order, the pair (A, B) is primitive if there exist nonnegative integers h and k such that the sum of all products formed by words consisting of h A's and k B's is a positive matrix. The exponent of the pair (A, B) is the smallest value of h + k over all such h and k. There is a natural correspondence between two-colored digraphs and nonnegative matrix pairs. In this paper, we consider the exponent of a especial nonnegative matrix pair whose associated digraph consists of two cycles.