The bandwidth sum of join and composition of graphs

Given a graph G, a proper labeling f of G is a one-to-one function . The bandwidth sum of a graph G, denoted by B_s(G), is defined by B_s(G)=min∑_uvE(G)|f(u)-f(v)|, where the minimum is taken for all proper labelings f of G. In this paper, we give some results for the bandwidth sum problem for the join of k graphs G₁,G₂,…,G_k, where each G_i is a path, cycle, complete graph, or union of isolated vertices. We also discuss the bandwidth sum for the composition of two graphs G and H, where G and H are path, cycle, or union of isolated vertices.