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

{ homomorphisms from graph x, y} ∈ E, then {f(x), f(y)} ∈ G into graph E . Let Hom H. The diamond product of (G, H) be the class of all

a graph G = (V, E) with a graph H = (V , E ) (denoted by G H) is a

graph defined by the vertex set V (G H) = Hom(G, H) and the edge set

E(G H) = {{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 m

m

n

n

+ nmmn vertices with diameter equal to two.