Cyclic property of iterative eccentrication of a 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 constructing iterative eccentric graphs, denoted by [Formula: see text] is called eccentrication. A graph [Formula: see text] is said to be [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 simple graph. The importance of this result lies in the fact that the enumeration of eccentrication of a graph reduces to a finite problem. Furthermore, the enumeration of a corresponding sequence of graph parameters such as chromatic number, domination number, independence number, minimum and maximum degree, etc., reduces to a finite problem.