Study on Harmonious Labeling of Symmetrical Trees of Diameter Four

A harmonious labeling on a graph G is an injection from the vertices of G to the group of integers modulo k, where k is the number of edges of G, that induces a bijection between the edges of G and the numbers modulo k by taking the edge label for an edge ab to be the sum of the labels of the two vertices a b(modk).In this paper we prove that the symmetrical trees of diameter four admit harmonious labeling. Harmonious graphs can be used in the study of error correcting codes and channel assignment problems.