Endomorphism monoid of diamond product of two common complete bipartite graphs

An endomorphism of a graph G = (V, E) is a mapping f : V → V such that for all x, y ∈ V if {x, y} ∈ E, then {f (x),f (y)}∈ E. Let End(G) be the class of all endomorphisms of graph G. The diamond product of graph G = (V, E) (denoted by G ◊ G) is a graph defined by the vertex set V (G ◊ G) = End(G) and the edge set E (G ◊ G) ={{f, g} ⊂ End(G)|{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 algebraic property of V (Km,n ◊ Km,n) = End(Km,n) after we have found that Km,n ◊ Km,n is also a complete bipartite graph on mmnn + nmmn vertices. The result shows that all of its vertices (endomorphisms) form a noncommutative monoid.