On a bipartite graph defined on groups
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866917861064179712 |
|---|---|
| author | Das, Shrabani Erfanian, Ahmad Nath, Rajat Kanti |
| author_facet | Das, Shrabani Erfanian, Ahmad Nath, Rajat Kanti |
| contents | Let $G$ be a group and $L(G)$ be the set of all subgroups of $G$. We introduce a bipartite graph $\mathcal{B}(G)$ on $G$ whose vertex set is the union of two sets $G \times G$ and $L(G)$, and two vertices $(a, b) \in G \times G$ and $H \in L(G)$ are adjacent if $H$ is generated by $a$ and $b$. We establish connections between $\mathcal{B}(G)$ and the generating graph of $G$. We also discuss about various graph parameters such as independence number, domination number, girth, diameter, matching number, clique number, irredundance number, domatic number and minimum size of a vertex cover of $\mathcal{B}(G)$. We obtain relations between $\mathcal{B}(G)$ and certain probabilities associated to finite groups. We also obtain expressions for various topological indices of $\mathcal{B}(G)$. Finally, we realize the structures of $\mathcal{B}(G)$ for the dihedral groups of order $2p$ and $2p^2$ and dicyclic groups of order $4p$ and $4p^2$ (where $p$ is any prime) including certain other small order groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_05494 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On a bipartite graph defined on groups Das, Shrabani Erfanian, Ahmad Nath, Rajat Kanti Group Theory Combinatorics Let $G$ be a group and $L(G)$ be the set of all subgroups of $G$. We introduce a bipartite graph $\mathcal{B}(G)$ on $G$ whose vertex set is the union of two sets $G \times G$ and $L(G)$, and two vertices $(a, b) \in G \times G$ and $H \in L(G)$ are adjacent if $H$ is generated by $a$ and $b$. We establish connections between $\mathcal{B}(G)$ and the generating graph of $G$. We also discuss about various graph parameters such as independence number, domination number, girth, diameter, matching number, clique number, irredundance number, domatic number and minimum size of a vertex cover of $\mathcal{B}(G)$. We obtain relations between $\mathcal{B}(G)$ and certain probabilities associated to finite groups. We also obtain expressions for various topological indices of $\mathcal{B}(G)$. Finally, we realize the structures of $\mathcal{B}(G)$ for the dihedral groups of order $2p$ and $2p^2$ and dicyclic groups of order $4p$ and $4p^2$ (where $p$ is any prime) including certain other small order groups. |
| title | On a bipartite graph defined on groups |
| topic | Group Theory Combinatorics |
| url | https://arxiv.org/abs/2412.05494 |