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Nonderogatory directed windmills. (English) Zbl 1189.05104

Summary: A directed graph G is nonderogatory [non-degenerated] if its adjacency matrix A is nonderogatory, i.e., the characteristic polynomial of A is equal to the minimal polynomial of A. Given integers r2y, h3, a directed windmill M h (r) is a directed graph obtained by coalescing r dicycles of length h 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]: M h (r) is nonderogatory if and only if r=2.

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.

05C50Graphs and linear algebra