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 f:V(G)→{1,2,…,|V(G)|}. The bandwidth sum of a graph G, denoted by Bs(G), is defined by Bs(G)=min∑uv∈E(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,…,Gk, where each Gi 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.