Assume that $G$ is a finite group. For every $a, b \in\mathbb N,$ we define a graph $\Gamma_{a,b}(G)$ whose vertices correspond to the elements of $G^a\cup G^b$ and in which two tuples $(x_1,\dots,x_a)$ and $(y_1,\dots,y_b)$ are adjacent if and only if $\langle x_1,\dots,x_a,y_1,\dots,y_b \rangle =G.$ We study several properties of these graphs (isolated vertices, loops, connectivity, diameter of the connected components) and we investigate the relations between their properties and the group structure, with the aim of understanding which information about $G$ are encoded by these graphs.

中文翻译：

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编码有限群的生成特性的图

假设$G$ 是一个有限群。对于每一个 $a, b \in\mathbb N,$ 我们定义一个图 $\Gamma_{a,b}(G)$ 其顶点对应于 $G^a\cup G^b$ 的元素，其中两个元组 $(x_1,\dots,x_a)$ 和 $(y_1,\dots,y_b)$ 相邻当且仅当 $\langle x_1,\dots,x_a,y_1,\dots,y_b \rangle =G.$我们研究了这些图的几个属性（孤立的顶点、环、连通性、连通分量的直径），并研究了它们的属性与群结构之间的关系，目的是了解这些图编码了关于 $G$ 的哪些信息.