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We consider the case in which mixed graphs (with both directed and undirectededges) are Cayley graphs of Abelian groups. In this case, some Moore boundswere derived for the maximum number of vertices that such graphs can attain. Wefirst show these bounds can be improved if we know more details about the orderof some elements of the generating set. Based on these improvements, we presentsome new families of mixed graphs. For every fixed value of the degree, thesefamilies have an asymptotically large number of vertices as the diameterincreases. In some cases, the results obtained are shown to be optimal.

An improved Moore bound and some new optimal families of mixed Abelian Cayley graphs