Let G be a group. The directed endomorphism graph, 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 b) if and only if there exists an endomorphism on G mapping a to b. The endomorphism graph, \, 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 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.
Ajith et al. (Sun,) studied this question.
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