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arXiv · 2609.00102

On the Generating Graph of Finite Abelian Groups

Kavita Samant·A. Satyanarayana Reddy
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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.

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Kavita Samant, A. Satyanarayana Reddy. 2026-08-31. On the Generating Graph of Finite Abelian Groups. https://arxiv.org/abs/2609.00102

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