Some results on λ-valuation of graphs involving complete bipartite graphs

Abstract

In this paper we show that a graph G obtained from a complete bipartite graph K m, n and a collection of q (⩽max\\s{ m, n\\s}) stars G i by joining the centre of G 1 to every vertex of K m, n and joining the centre of G i to a vertex (not the centre) of G i+1 ( i=1,2,…, q−1) is strongly harmonious. We also prove that a graph obtained from a collection of t complete bipartite graphs K m i , n i with bipartition ( X i , Y i ) by joining exactly one member in Y i with a member in X i+1 ( i=1,2,…, t−1) is strongly c-elegant. The windmill graph K ( n) 2,2 of n complete bipartite graphs K 2,2 with a common vertex is also shown to be harmonious for all n⩾2.