On Co-complete k–partite Graph Valued Functions

Abstract

A co-complete k -partite graph G = (V1,V2, … ,Vk, E), k ≥ 2 is a graph with the smallest number k of disjoint parts in which any pair of vertices in the same part is at a distance two. The number of parts in co-complete k -partite graph G is denoted by k (G). In this paper, we investigate k() , k(L(G)), k(M(G)) and k(T(G)), where , L(G) , M(G) and T(G) are complement graph, line graph, middle graph and total graph, respectively of some standard graphs. We also discuss the relationship between them. Each set in the partition has subpartitions such that each set in the subpartition induces K1 or any two vertices in this subpartition are at distance two and this result has significance in providing a stable network.