Cyclic property of iterative eccentrication of trees

A tree graph is an acyclic graph. The eccentric graph of a graph [Formula: see text], denoted by [Formula: see text] is a derived graph with the vertex set same as that of [Formula: see text] and two vertices in [Formula: see text] are adjacent if one of them is an eccentric vertex of the other. The process of finding eccentric graph of a graph is called eccentrication and that of constructing iterative eccentric graphs, denoted by [Formula: see text], is called iterative eccentrication. A graph [Formula: see text] is said to be [Formula: see text]-cyclic[Formula: see text] if [Formula: see text] are the only non-isomorphic graphs, and the graph [Formula: see text] is isomorphic to [Formula: see text]. In this paper, we prove the existence of an [Formula: see text]-cycle for any tree graph on [Formula: see text] vertices. We also obtain some important results on eccentric graphs of trees. Then, we present a conjecture on the cyclic property of eccentrication of a general graph [Formula: see text]. Finally, an analogy between the concept of [Formula: see text] for a graph and the dichotomy of the Riemann sphere into Fatou sets and Julia sets is presented. We also state some open problems in the area.