Hamiltonian cycles for finite Weyl groupoids

Let Γ (W) be the Cayley graph of a finite Weyl groupoid . In this paper, we show an existence of a Hamiltonian cycle of Γ (W) for any . We exactly draw a Hamiltonian cycle of Γ (W) for any (resp. some) irreducible of rank three (resp. four). Moreover for the irreducible of rank three, we give a second largest eigenvalue of the adjacency matrix of Γ (W), and know if Γ(W) is a bipartite Ramanujan graph or not.