On the Generating Graph of Finite Abelian Groups

Abstract

The generating graph $\Gamma (G)$ 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 $\Gamma (G)$ 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 $\Gamma (G)\cong \Gamma (H)$. Furthermore, we compute the spectra of the adjacency and Laplacian matrices of these graphs.

23 pages