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A class of directed graph, when their minimal polynomial of the adjacency matrix matches the characteristic polynomial

We consider an unweighted directed simple graph, $G$, with a Hamiltonian cycle. Q. Assume that the adjacency matrix of $G$ is non-singular. Do the characteristic and minimal polynomials of the ...
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Are there any necessary conditions of the existence of a Hamiltonian cycle on directed graphs

I'm trying to prove that one concrete directed graph has no Hamiltonian cycle, but didn't seem to find any relevant theorems
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