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arXiv · 2503.00759

Endomorphism and Automorphism Graphs

Midhuna V Ajith·Mainak Ghosh·Aparna Lakshmanan S
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Abstract

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.

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Midhuna V Ajith, Mainak Ghosh, Aparna Lakshmanan S. 2025-03-02. Endomorphism and Automorphism Graphs. https://arxiv.org/abs/2503.00759

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