1. Introduction
Consider a simple finite graph. Terms and terminologies have been referred from the book [1]. A graph G’ with graceful labeling is given by a function g. It contains an edge with injective function g assigned from V(G’) to {0,1,2,…,r}. When all the edges ‘ab’ is reserved to be the edge label |g(a)− g(b)|, at that point the subsequent edge names are particular. The graceful labeling accepted by a graph is known as a graceful graph. Some classes of trees using graceful labeling were studied by A. Rosa 1966 [2]-paths and caterpillars, G. Sethuraman et al 2009 [3]-attaching caterpillars with other caterpillars, I. Cahit 2002 [4]-canonic labeling technique, R.E. Aldred et al 1998 [5] -lobsters, Jesintha et al 2011 [6]- constructed certain infinite families of graceful Acharya 1982 [7], A. Gallian 2019 [8]-surveyed on graceful tree conjecture [9-22]. A star graph can be defined as the complete bipartite graph 𝐾1,r containing r + 1 vertices and r edges. A simple graph can be defined as a graph that does not contain a self-loop or parallel edges. A path 𝑃𝑛 is acquired by signing up for 𝑢𝑖 to the consecutive vertices 𝑢𝑖+1 for 1 ≤ 𝑖 ≤ 𝑛 − 1.
In mathematical models, In a wide variety of mathematical models, Labeled graphs have played a remarkable usage. In the application of designing X-Ray crystallography, formulating an addressing system using a communication network, determining the layouts of a perfect circuit, problems based on the addition of numbers, etc. are used.