Endomorphism and automorphism graphs of finite groups

Abstract: Let $G$ be a group. The directed endomorphism graph, $\dend(G)$ of $G$ is a directed graph with vertex set $G$ and there is a directed edge from the vertex $a$ to the vertex $b$ if $a \neq b$ and there exists an endomorphism on $G$ mapping $a$ to $b$. The endomorphism graph, $\uend(G)$ is the corresponding undirected simple graph. The automorphism graph of $G$ is similarly defined for automorphisms: it is a disjoint union of complete graphs on the orbits of $\Aut(G)$. The endomorphism digraph is a special case of a digraph associated with a transformation monoid, and we begin by introducing this. 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, as well as computing these graphs for some special groups. We conclude with examples showing that things are not always simple.