The Number of Labelled k-Arch Graphs
نویسنده
چکیده
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 ≥ 1, is (nk)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 nn−2 that counts the number of labelled trees.
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