发表机构
Shiv Nadar Institution of Eminence（Shiv Nadar卓越学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究非循环有限阿贝尔群的生成图结构，确定生成对集合，刻画其正则性、孤立顶点子群性等，给出两群生成图同构的充要条件并计算两类矩阵的谱。
AI 中文摘要
群G的生成图Γ(G)是顶点集为G的图，其中两个不同顶点相邻当且仅当它们生成G。本文系统研究非循环有限阿贝尔群的生成图结构，确定所有生成对的集合，给出若干结构刻画，特别是确定Γ(G)正则的条件、孤立顶点构成子群的情形，建立两个非同构有限阿贝尔群G和H满足Γ(G)≅Γ(H)的充要条件，还计算了这类图的邻接矩阵与拉普拉斯矩阵的谱。
英文摘要
The generating graph $Γ(G)$ of a group $G$ is the graph whose vertex set is $G$, where two distinct vertices are adjacent if and only if they generate $G$. In this paper, we systematically study the structure of generating graphs of finite abelian groups (non-cyclic) and determine the set of all generating pairs. Moreover, we give some structural characterizations, in particular, we determine conditions under which $Γ(G)$ is regular, characterize when the isolated vertices form a subgroup, and establish necessary and sufficient conditions for two non-isomorphic finite abelian groups $G$ and $H$ to satisfy $Γ(G)\cong Γ(H)$. Furthermore, we compute the spectra of the adjacency and Laplacian matrices of these graphs.
Comments23 pages
