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Nonderogatory directed windmills

Keywords: nonderogatory matrix, characteristic polynomial of directedgraphs, directed windmills.

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Abstract:

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.

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