The composition of two graphs G and H, written G[H], is the graph with vertex setV (G) V(H) and with (u 1 ,v 1 ) adjacent to (u 2 ,v 2 ) if either u 1 is adjacent to u 2 in G oru 1 = u 2 and v 1 is adjacent to v 2 in H. In this paper, we investigate the bandwidth problem for the composition of two graphs and obtain the bandwidth for several classes of graphs of the formsK n [G] and K 1 , n [G]. We also provide optimal numberings which solve the graph edgesum problem for K n [P m ] and K n [C m ].