Cyclic property of iterative eccentrication of a graph.

The eccentric graph of a graph G , denoted by G e , is a derived graph with the vertex set same as that of G and two vertices in G e are adjacent if one of them is an eccentric vertex of the other. The process of constructing iterative eccentric graphs, denoted by G e k is called eccentrication. A graph G is said to be ℰ − cyclic (t , l) if G , G e , G e 2 , ... , G e k , G e k + 1 , ... , G e k + l are the only non-isomorphic graphs, and the graph G e k + l + 1 is isomorphic to G e k . In this paper, we prove the existence of an ℰ -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. [ABSTRACT FROM AUTHOR]