(k,d)-Balanced of Uniform k-Distant Trees

A k-distant tree consists of a main path called the “spine”, such that each vertex on the spine is joined by an edge to at most one path on k-vertices. Those paths are called “tails” (i.e. each tail must be incident with a vertex on the spine). When every vertex on the spine has exactly one incident tail of length k, we call the tree a uniform k-distant tree. In this paper, we show that all uniform k-distant trees are graceful, skolem graceful, odd graceful, k-graceful, (k, d)-graceful and (k, d)-balanced.