In 1966, Rosa introduced, among others, α- and β-labelings as tools to solve the isomorphic decomposition problem of the complete graph. Ten years later, Sheppard calculated the number of α- and β-labeled graphs with n edges. In this paper we use an extended version of the adjacency matrix of a graph to determine the number of gracefully labeled graphs with n edges; furthermore, we also calculate the number of them with m vertices for every suitable value of m. In addition, we use this technique to determine the number of labeled graphs for other types of labelings as the harmonious, felicitous, and elegant.