Spectral Bounds of the Generating Graph of $\mathbb{Z}_n.$
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866909458607636480 |
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| author | Samant, Kavita Reddy, A. Satyanarayana |
| author_facet | Samant, Kavita Reddy, A. Satyanarayana |
| contents | Let $G$ be a group. A group is said to be $k$-generated if it can be generated by its $k$ elements. A generating set of $G$ is called a minimal generating set if no proper subset of it generates $G.$ A minimal generating set of a group can have different sizes. The generating graph $Γ(G)$ of a group $G$ is defined as a graph with the vertex set $G$, where two distinct vertices are adjacent if they together generate $G.$ This graph is particularly useful when studying 2-generated groups. In this context, consider the group $G = \mathbb{Z}_n$, the integers modulo $n.$ In this paper, we explore various graph-theoretic properties of the generating graph $Γ(\mathbb{Z}_n)$ and investigate the spectra of its adjacency and Laplacian matrices. Additionally, we explicitly determine the set of all possible minimal generating sets of $\mathbb{Z}_n$ of size $k.$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_09771 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Spectral Bounds of the Generating Graph of $\mathbb{Z}_n.$ Samant, Kavita Reddy, A. Satyanarayana Combinatorics Group Theory 05C25, 05C50, 20D60 Let $G$ be a group. A group is said to be $k$-generated if it can be generated by its $k$ elements. A generating set of $G$ is called a minimal generating set if no proper subset of it generates $G.$ A minimal generating set of a group can have different sizes. The generating graph $Γ(G)$ of a group $G$ is defined as a graph with the vertex set $G$, where two distinct vertices are adjacent if they together generate $G.$ This graph is particularly useful when studying 2-generated groups. In this context, consider the group $G = \mathbb{Z}_n$, the integers modulo $n.$ In this paper, we explore various graph-theoretic properties of the generating graph $Γ(\mathbb{Z}_n)$ and investigate the spectra of its adjacency and Laplacian matrices. Additionally, we explicitly determine the set of all possible minimal generating sets of $\mathbb{Z}_n$ of size $k.$ |
| title | Spectral Bounds of the Generating Graph of $\mathbb{Z}_n.$ |
| topic | Combinatorics Group Theory 05C25, 05C50, 20D60 |
| url | https://arxiv.org/abs/2501.09771 |