编码有限群生成性质的图
Graphs encoding the generating properties of a finite group
摘要：
假设$G$是一个有限群。对于每个$a, b \in\mathbb N,$，我们定义了一个图$Γ_{a,b}(G)$，它的顶点对应于$G^a\cup G^b$的元素，并且其中两个元组$(x_1,\dots,x_a)$和$(y_1,\dots,y_b)$相邻当且仅当$\langle x_1,\dots,x_a,y_1,\dots,y_b \rangle =G.$。（孤立点，回路，连通性，连通分支的直径），并研究了它们的性质与群结构之间的关系，目的是理解关于$G$的哪些信息由这些图编码。
英文摘要：
Assume that $G$ is a finite group. For every $a, b \in\mathbb N,$ we define a graph $Γ_{a,b}(G)$ 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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
