Cyclic property of iterative eccentrication of a graph
Abstract
The eccentric graph of a graph GG, denoted by GeGe, is a derived graph with the vertex set same as that of GG and two vertices in GeGe are adjacent if one of them is an eccentric vertex of the other. The process of constructing iterative eccentric graphs, denoted by GekGek is called eccentrication. A graph GG is said to be ℰ−cyclic(t,l) if G,Ge,Ge2,…,Gek,Gek+1,…,Gek+l are the only non-isomorphic graphs, and the graph Gek+l+1 is isomorphic to Gek. 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.
Communicated by Xueliang Li
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