The number of labeled k-dimensional trees

AbstractThe graphs known as trees have natural analogues in higher dimensional simplicial complexes. As an extension of Cayley's formula nn−2, the number of these k-dimensional trees on n-labeled vertices is shown to be (nk) (kn−k2+1)n−k−2

On the honesty of graph complements

AbstractA graph is called honest if its edge-integrity equals its order. It is shown in this paper that except for the path of length 3, every graph that is not honest has an honest complement. This result is extended to complements of products and applied to the Nordhaus-Gaddum theory for edge- integrity