A graph whose full automorphism group is isomorphic to a finite group G is called a G-graph, and we let \(\alpha (G)\) denote the minimal number of vertices among all G-graphs. The value of \(\alpha (G)\) has been established for numerous infinite families of groups. In this article, we expand upon the subject matter of vertex-minimal G-graphs by computing the value of \(\alpha (G)\) when G is isomorphic to either a quasi-dihedral group or a quasi-abelian group. These results completely establish the value of \(\alpha (G)\) when G is a member of one of the six infinite families of 2-groups that contain a cyclic subgroup of index 2.

中文翻译：

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具有nonabelian $$ {\ mathbf {2}} $$ 2-组对称的顶点最小图

完全自同构群与有限群G同构的图称为G-图，我们令\（\ alpha（G）\）表示所有G图中的最小顶点数。的值\（\α（G）\）已建立组的众多的无限族。在本文中，我们通过计算当G与拟二面体组或拟阿贝尔组同构时的\（\ alpha（G）\）的值来扩展顶点最小G图的主题。当G时，这些结果完全确定\（\ alpha（G）\）的值 是包含2的循环子组的6个2组无限族之一的成员。