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PreprintCC BY 4.0arXiv

On the Generating Graph of Finite Abelian Groups

  • Kavita Samant
  • A. Satyanarayana Reddy
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Abstract

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)≅ Γ(H). Furthermore, we compute the spectra of the adjacency and Laplacian matrices of these graphs.

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Cite

BibTeXRIS
BibTeX
@misc{samant2026generating,
  title = {{On the Generating Graph of Finite Abelian Groups}},
  author = {Kavita Samant and A. Satyanarayana Reddy},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.00102},
  eprint = {2609.00102},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.00102},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - On the Generating Graph of Finite Abelian Groups
AU  - Kavita Samant
AU  - A. Satyanarayana Reddy
PY  - 2026
DA  - 2026/09/01
PB  - arXiv
DO  - 10.48550/arxiv.2609.00102
UR  - https://arxiv.org/abs/2609.00102
AB  - 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)≅ Γ(H). Furthermore, we compute the spectra of the adjacency and Laplacian matrices of these graphs.
ER  -