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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 \(n&gt;1\), the graphs \(G(E_{n})\) and \(G_{E_{n}}\) are Hamiltonian.

UNIT AND UNITARY CAYLEY GRAPHS FOR THE RING OF EISENSTEIN INTEGERS MODULO \(n\)