DIAMOND PRODUCT OF TWO COMMON COMPLETE BIPARTITE GRAPHS

Abstract

A homomorphism of a graph G =(V,E) into a graph H =( V ,E ) is a mapping f : V −→ V which preserves edges: for all x,y ∈ V , if {x,y}∈E, then{f(x),f(y)}∈E. LetHom(G,H) be the class of all homomorphisms from graph G into graph H. The diamond product of a graphG =(V,E) with a graph H =(V ,E ) (denoted by GH) is a graph defined by the vertex set V (GH)=Hom(G,H) and the edge set E(GH)={{f,g}⊂Hom(G,H)|{f(x),g(x)}∈E for all x ∈ V}. Let Km,n be a complete bipartite graph on m+n vertices. This research aims to study the diamond product of two common complete bipartite graphs Km,n. We find that the resulting graph is also a complete bipartite graph on mmnn + nmmn vertices with diameter equal to two.