Unit and unitary Cayley graphs for the ring of Eisenstein integers modulo $n$

Let ${E}_{n}$ be the ring of Eisenstein integers modulo $n$. We denote by $G({E}_{n})$ and $G_{{E}_{n}}$, the unit graph and the unitary Cayley graph of ${E}_{n}$, 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 $G(E_{n})$ and $G_{E_{n}}$ are Hamiltonian.