The fundamental graph labelings as well as the harmonious labelling are defined. Labeling and cordial labelling are also separated into graceful and harmonious categories. This labelling is used throughout the thesis. The conjecture sparked a lot of interest in graph labelings in the mid 1960s. The accepted Ringel hypothesis is that all trees are graceful, but excessively unsettled. Rosa introduced valuation and other labelings as a tool for decomposing complete graphs in his seminal paper. The term most commonly used in harmonious extensively studied inevaluation, which was later called graceful labelling. In the field of graph labelling, variations of graceful and harmonious labelling, specifically appraisal, elegant and cordial labelling, have been introduced with various motivations. More than 600 papers have been written on this subject over the last four decades. This demonstrates the field's rapid expansion. However, the fundamental understanding that the description of graceful and previously labelled graph appears to be one of the most complex and difficult problems in graph theory. Indeed, approval of the simplest labelled graph, cordial graph, is an NP-complete problem.