Hamiltonian Cycles for Finite Weyl Groupoids

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