Geodesic
The number of labelled \(k\)-arch graphs
Journal of integer sequences, Tome 7 (2004) no. 3
In this note, we deal with $k$-arch graphs, a generalization of trees, which contain $k$-trees as a subclass. We show that the number of vertex-labelled $k$-arch graphs with $n$ vertices, for a fixed integer $k\geq 1$, is ${n\choose k}^{n-k-1}$. As far as we know, this is a new integer sequence. We establish this result with a one-to-one correspondence relating $k$-arch graphs and words whose letters are $k$-subsets of the vertex set. This bijection generalises the well-known Prüfer code for trees. We also recover Cayley's formula $n^{n-2}$ that counts the number of labelled trees.
Classification : 05A15, 05C30, 05A10
Keywords: k-arch graphs, pr$\ddot $ufer code, generalization of trees 
@article{JIS_2004__7_3_a3,
     author = {Lamathe,  C\'edric},
     title = {The number of labelled \(k\)-arch graphs},
     journal = {Journal of integer sequences},
     year = {2004},
     volume = {7},
     number = {3},
     zbl = {1065.05052},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JIS_2004__7_3_a3/}
}
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Lamathe,  Cédric. The number of labelled \(k\)-arch graphs. Journal of integer sequences, Tome 7 (2004) no. 3. http://geodesic.mathdoc.fr/item/JIS_2004__7_3_a3/