Formulations and Solutions for Graphs of K-Ary Trees using Various Mathematical Algorithms to Determine Graceful Labeling Considering Binary, Ternary, Quaternary and 5-Ary Symmetrical Trees

Abstract

The field of graph theory plays a vital role in various fields related to mathematical modeling. Graph labeling is a assignment of integers either to the vertices or edges or both subject to certain conditions. One of the specific and effective ways of graph labeling is called Graceful labeling. This is a technique which we label a graph in a certain way so that the edges and vertex labels are distinct. In the tree family of symmetric graphs, a k-ary tree is a rooted tree in which each vertex has at-most k children. The following paper is devoted to provide all possible algorithms which are useful for the graceful labeling of quaternary trees or all K –ary trees. Solutions become more effective with illustration figures. Here we use numerous algorithms of different patterns using various mathematical combinations to try to gracefully label ternary and quaternary trees. At the end of the research work, we are finally able to label the quaternary tree using these patterns in a graceful manner.