Endomorphism and Automorphism Graphs

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 $b$’ $(a\mathrm{\ne }b)$ 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 $b$’ $(a\mathrm{\ne }b)$ 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.