Proper Colourings in r -Regular Inverse Sum Indeg Index Graphs

Abstract

Let G(V,E) be a simple undirected graph. The Inverse sum indeg index of a graph G is defined as $$ISI(G) = \sum_{r,s{\in}E(G)} {drds \over dr + ds}$$ where ds is the degree of the vertex s in G. The new idea of proper colourings in the Inverse sum indeg index graph has been proposed in this chapter. Some families of graphs such as cycle, generalized Petersen graph and complete graph which satisfies the condition of r–regular Inverse sum indeg index. The inequalities for the chromatic number related to Inverse sum indeg index are developed.