Summary: A directed graph is nonderogatory [non-degenerated] if its adjacency matrix is nonderogatory, i.e., the characteristic polynomial of is equal to the minimal polynomial of . Given integers , , a directed windmill is a directed graph obtained by coalescing dicycles of length in one vertex. In this article we resolve a conjecture proposed by C. S. Gan and V. C. Koo [On annihilating uniqueness of directed windmills, Proc. of ATCM 2002, Melaka, Malaysia]: is nonderogatory if and only if .
Editorial remark: This paper is a translation of the author’s article [Notas Mat. 2, No. 2, 69-74 (2006; Zbl 1146.05305)] from Spanish, without mentioning the original publication.