The Difference Subgroup Graph of a Finite Group
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866909890620948480 |
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| author | Das, Angsuman Mandal, Arnab Sarkar, Labani |
| author_facet | Das, Angsuman Mandal, Arnab Sarkar, Labani |
| contents | The \emph{difference subgroup graph} $D(G)$ of a finite group $G$ is defined as the graph whose vertices are the non-trivial proper subgroups of $G$, with two distinct vertices $H$ and $K$ adjacent if and only if $\langle H, K \rangle = G$ but $HK \ne G$. This graph arises naturally as the difference between the join graph $Δ(G)$ and the comaximal subgroup graph $Γ(G)$. In this paper, we initiate a systematic study of $D(G)$ and its reduced version $D^*(G)$, obtained by removing isolated vertices.
We establish several fundamental structural properties of these graphs, including conditions for connectivity, forbidden subgraph characterizations, and the relationship between graph parameters - such as independence number, clique number, and girth - and the solvability or nilpotency of the underlying group.
The paper concludes with a discussion of open problems and potential directions for future research. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_04411 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Difference Subgroup Graph of a Finite Group Das, Angsuman Mandal, Arnab Sarkar, Labani Group Theory Combinatorics 05C25 The \emph{difference subgroup graph} $D(G)$ of a finite group $G$ is defined as the graph whose vertices are the non-trivial proper subgroups of $G$, with two distinct vertices $H$ and $K$ adjacent if and only if $\langle H, K \rangle = G$ but $HK \ne G$. This graph arises naturally as the difference between the join graph $Δ(G)$ and the comaximal subgroup graph $Γ(G)$. In this paper, we initiate a systematic study of $D(G)$ and its reduced version $D^*(G)$, obtained by removing isolated vertices. We establish several fundamental structural properties of these graphs, including conditions for connectivity, forbidden subgraph characterizations, and the relationship between graph parameters - such as independence number, clique number, and girth - and the solvability or nilpotency of the underlying group. The paper concludes with a discussion of open problems and potential directions for future research. |
| title | The Difference Subgroup Graph of a Finite Group |
| topic | Group Theory Combinatorics 05C25 |
| url | https://arxiv.org/abs/2511.04411 |