Classification of n × (n − 1) Incidence Matrices of Gracefully Labeled Simple Graphs

This paper presents the distribution of simple graphs with $n$ vertices and $n-1$ edges that admit graceful labeling. This is achieved through a classification table, which considers the graph's diameter (or longest cycle) and the number of isolated vertices. We determine the degree sequences by utilizing the incidence matrices of the labeled graphs mentioned above and subsequently arrange the collection on this classification table. Using the table, we investigate the impact of degree sequences on the diameter (or the longest cycle) value and the existence of isolated vertices.