Unit and Unitary Cayley Graphs for the Ring of Eisenstein Integers Modulo N

Let En be the ring of Eisenstein integers modulo n. We denote by G(En) and GEn, the unit graph and the unitary Cayley graph of En, respectively. In this paper, we obtain the value of the diameter, the girth, the clique number and the chromatic number of these graphs. We also prove that for each n>1, the graphs G(En) and GEn are Hamiltonian.