Endomorphism and Automorphism Graphs

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.