AI 中文总结
本文解决了几乎混合摩尔图的全正则性开放问题，证明了3个直径至少为3的已知几乎混合摩尔图是仅有的此类混合图，相关成果可应用于网络设计领域。
AI 中文摘要
度/直径问题是指，给定直径和最大顶点度时，求图的最大顶点数，该问题已被广泛研究，并衍生出混合图（同时包含无向边和有向弧的图）的相关变体，其中对任意顶点的最大有向出度施加了额外约束。这两类问题均适用于网络设计。统计从给定顶点出发各距离处可能的顶点数，可得到满足度和直径约束的混合图顶点数的一个界（混合摩尔界）。本文中，我们解决了Tuite和Erskine提出的关于顶点数比混合摩尔界小1的混合图（几乎混合摩尔图）的全正则性的开放问题。我们利用该结果证明，已知的3个直径至少为3的几乎混合摩尔图是仅有的此类混合图。
英文摘要
The degree/diameter problem asks for the largest order of a graph with a given diameter and maximum vertex degree. This has been widely studied and given rise to a recent variation for mixed graphs (graphs with both undirected edges and directed arcs), where an additional bound is placed on the maximum directed out-degree of any vertex. Both problems have applications to network design. Counting the possible number of vertices at each distance from a given vertex gives a bound on the order of a mixed graph satisfying the degree and diameter constraints (the mixed Moore bound). In this paper, we settle an open problem posed by Tuite and Erskine concerning the total regularity of mixed graphs whose order is one less than the mixed Moore bound (almost mixed Moore graphs). We use this result to show that the three known almost mixed Moore graphs of diameter at least three are the only such mixed graphs.
