Geodesic
Enumerating families of labeled graphs
Journal of integer sequences, Tome 18 (2015) no. 1
In 1966, Rosa introduced, among others, $\alpha $- and $\beta $-labelings as tools to solve the isomorphic decomposition problem of the complete graph. Ten years later, Sheppard calculated the number of $\alpha $- and $\beta $-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.
Classification : 05C30, 05C78
Keywords: graceful graph, $\alpha $-graph, enumeration, graceful triangle 
@article{JIS_2015__18_1_a7,
     author = {Barrientos,  Christian and Minion,  Sarah},
     title = {Enumerating families of labeled graphs},
     journal = {Journal of integer sequences},
     year = {2015},
     volume = {18},
     number = {1},
     zbl = {1309.05098},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JIS_2015__18_1_a7/}
}
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Barrientos,  Christian; Minion,  Sarah. Enumerating families of labeled graphs. Journal of integer sequences, Tome 18 (2015) no. 1. http://geodesic.mathdoc.fr/item/JIS_2015__18_1_a7/