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| Format: | Preprint |
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2025
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| Online-Zugang: | https://arxiv.org/abs/2503.00759 |
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| _version_ | 1866911317147779072 |
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| author | Ajith, Midhuna V Ghosh, Mainak S, Aparna Lakshmanan |
| author_facet | Ajith, Midhuna V Ghosh, Mainak S, Aparna Lakshmanan |
| contents | Let $G$ be a group. The directed endomorphism graph, \dend of $G$ is a directed graph with vertex set $G$ and there is a directed edge from the vertex `$a$' to the vertex `$\, b$' $(a \neq b) $ if and only if there exists an endomorphism on $G$ mapping $a$ to $b$. The endomorphism graph, \uend $\,$ of $G$ is the corresponding undirected simple graph. The automorphism graph, ${Auto}(G)$ of $G$ is an undirected graph with vertex set $G$ and there is an edge from the vertex `$a$' to the vertex `$\,b$' $(a \neq b) $ if and only if there exists an automorphism on $G$ mapping $a$ to $b$. We have explored graph theoretic properties like size, planarity, girth etc. and tried finding out for which types of groups these graphs are complete, diconnected, trees, bipartite and so on. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_00759 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Endomorphism and Automorphism Graphs Ajith, Midhuna V Ghosh, Mainak S, Aparna Lakshmanan Combinatorics 05C20, 05C25, 08A35 Let $G$ be a group. The directed endomorphism graph, \dend of $G$ is a directed graph with vertex set $G$ and there is a directed edge from the vertex `$a$' to the vertex `$\, b$' $(a \neq b) $ if and only if there exists an endomorphism on $G$ mapping $a$ to $b$. The endomorphism graph, \uend $\,$ of $G$ is the corresponding undirected simple graph. The automorphism graph, ${Auto}(G)$ of $G$ is an undirected graph with vertex set $G$ and there is an edge from the vertex `$a$' to the vertex `$\,b$' $(a \neq b) $ if and only if there exists an automorphism on $G$ mapping $a$ to $b$. We have explored graph theoretic properties like size, planarity, girth etc. and tried finding out for which types of groups these graphs are complete, diconnected, trees, bipartite and so on. |
| title | Endomorphism and Automorphism Graphs |
| topic | Combinatorics 05C20, 05C25, 08A35 |
| url | https://arxiv.org/abs/2503.00759 |