This paper gives a graceful labeling to some new classes of diameter six and diameter seven trees using the concept of joining isomorphic copies of a tree (Koh and Tan, 1979; and Chen Wen-Chin et al., 1997).

The concept of graceful labeling originated as a means of attacking the famous and unsolved conjecture of Ringel (1964) related to the decomposition of complete graphs. The earliest name of graceful labeling is bvaluation (Rosa, 1968). Golomb (1972) subsequently called bvaluation as graceful labeling which is now the popular term. Later on several applications of graceful graphs were unearthed such as communication network addressing, X-ray crystallography, coding theory, rulers, radar and missile guidance, constrained satisfactory problems (Bloom and Golomb, 1977; and Bloom and Golomb, 1978), radio astronomy (Bermond et al., 1979; and Bloom, 1989), and oberwolfach problems (Gvozdjak, 2004).

One can notice that if we merge centers of two or more diameter four trees with the pendant vertices of a star we get a diameter six tree. Further, if we merge the centers of two or more stars with the pendant vertices of a diameter four tree which are at a distant two from the center, we get a diameter six tree.When we create an edge between the centers of two diameter six trees, a diameter seven tree is formed. Similarly, when we create an edge between one of the centers of a diameter five tree with one of the centers of another diameter five tree, a diameter seven tree is formed.

Till now it is known that all trees up to diameter five are graceful (see Bloom, 1989; Zhao, 1989; Cahit, 1990; Jin et al., 1993; Hrnciar Pavel and Havier Alfonz, 2001; and Gallian, 2003). Banana trees described in (Bhat-Nayak and Deshmukh, 1996) are the only diameter six trees which are known to be graceful. Further no effort has been made so far to give graceful labeling to the diameter seven trees. Motivated joining of m isomorphic copies of a graceful tree (Koh and Tan, 1979) by joining the isomorphic images of the vertex whose label is g, where q is the number of edges of the tree, merging isomorphic images of a vertex of a graceful tree with each vertex of another graceful tree (Koh and Tan, 1979), we attempt to give graceful labeling to some classes of diameter six trees. Further, joining isomorphic graceful trees by creating an edge between isomorphic images of a vertex (Chen Wen-Chin et al., 1997) we give graceful labeling to some classes of diameter seven trees.

Graceful labeling, Diameter four trees, Diameter six trees, Diameter seven trees, X-ray crystallography, Isomorphic images, Isomorphic graceful trees, Isomorphic copies, Graceful tree conjucture.