Endomorphism and Automorphism Graphs

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 `$b$' $(a\ne b)$ 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 `$b$' $(a\ne b)$ 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.

An updated version of the paper with one more coauthor is uploaded in arXiv by myself as arXiv:2511.15602