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

中文翻译：

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改进的摩尔界和一些新的混合阿贝尔凯莱图最优族

我们考虑混合图（具有有向边和无向边）是阿贝尔群的凯莱图的情况。在这种情况下，对于此类图可以达到的最大顶点数，推导出了一些摩尔边界。我们首先表明，如果我们了解有关生成集某些元素顺序的更多详细信息，则可以改进这些边界。基于这些改进，我们提出了一些新的混合图系列。对于度数的每个固定值，随着直径的增加，这些族具有渐近大量的顶点。在某些情况下，获得的结果被证明是最佳的。