Nonderogatory directed windmills

a directed graph g is nonderogatory if its adjacency matrix a is nonderogatory, i.e., the characteristic polynomial of a is equal to the minimal polynomial of a. given integers r≥ 2 and h≥ 3, a directed windmill mh(r) is a directed graph obtained by coalescing r dicycles of length h in one vertex. in this article we solve a conjecture proposed by gan and koo ([3]): mh(r) is nonderogatory if and only if r=2.