In this article we discuss about the topic under graph theory such as some basic definition, graceful graphs, Union of two graphs on classes of trees, pendant graceful graphs on edge labeling, Edge odd gracefulness graphs. Graph theory is the representation of an object in a set by graphically, which are associated by links and it is essentially concentrated in Mathematics as well as Computer Science. Graphs can be plotted by using vertices and edges. It is additionally used to examine the particles in physics and chemistry. In material physics we can study the atomic structures simulated by the structure of three dimensional. Also we have so many theorems in graph theory based on the coloring of graph, like edge coloring, vertex coloring. We can also study the path, walk, shortest path, Hamiltonian path etc. Nowadays Graph theory also used in Google map also to find the shortest distance between arrival and destination place.