Bandwidth of the composition of two graphs

The bandwidth B(G) of a graph G is the minimum of the quantity max{|f(x) − f(y)| : xy∈ E(G)} taken over all proper numberings f of G. The composition of two graphs G and H , written as G[H ], is the graph with vertex set V (G) × V (H) and with (u1; v1) is adjacent to (u2; v2) if either u1 is adjacent to u2 in G or u1 =u2 and v1 is adjacent to v2 in H . In this paper, we investigate the bandwidth of the composition of two graphs. Let G be a connected graph. We denote the diameter of G by D(G). For two distinct vertices x; y∈V (G), we de8ne wG(x; y) as the maximum number of internally vertex-disjoint (x; y)-paths whose lengths are the distance between x and y. We de8ne w(G) as the minimum of wG(x; y) over all pairs of vertices x; y of G with the distance between x and y is equal to D(G). Let G be a non-complete connected graph and let H be any graph. Among other results, we prove that if |V (G)|=B(G)D(G)−w(G) + 2, then B(G[H ]) = (B(G) + 1)|V (H)| − 1. Moreover, we show that this result determines the bandwidth of the composition of some classes of graphs composed with any graph. c © 2003 Elsevier B.V. All rights reserved.