Geodesic
On the number of labeled \(k\)-arch graphs
Journal of integer sequences, Tome 10 (2007) no. 7
In this paper we deal with $k$-arch graphs, a superclass of trees and $k$-trees. We give a recursive function counting the number of labeled $k$-arch graphs. Our result relies on a generalization of the well-known Prüfer code for labeled trees. In order to guarantee the generalized code to be a bijection, we characterize the valid code strings.
Classification : 05A15, 05C30, 05A10
Keywords: k-arch graphs, trees, k-trees, coding, pr$\ddot $ufer code, Cayley's formula 
@article{JIS_2007__10_7_a5,
     author = {Caminiti,  Saverio and Fusco,  Emanuele G.},
     title = {On the number of labeled \(k\)-arch graphs},
     journal = {Journal of integer sequences},
     year = {2007},
     volume = {10},
     number = {7},
     zbl = {1144.05035},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JIS_2007__10_7_a5/}
}
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Caminiti,  Saverio; Fusco,  Emanuele G. On the number of labeled \(k\)-arch graphs. Journal of integer sequences, Tome 10 (2007) no. 7. http://geodesic.mathdoc.fr/item/JIS_2007__10_7_a5/