  In this paper, we study the joint behaviour of the degree, depth and label of
and graph distance between high-degree vertices in the random recursive tree.
We generalise the results obtained by Eslava and extend these to include the
labels of and graph distance between high-degree vertices. The analysis of both
these two properties of high-degree vertices is novel, in particular in
relation to the behaviour of the depth of such vertices. In passing, we also
obtain results for the joint behaviour of the degree and depth of and graph
distance between any fixed number of vertices with a prescribed label. This
combines several isolated results on the degree and depth of and graph distance
between vertices with a prescribed label already present in the literature.
Furthermore, we extend these results to hold jointly for any number of fixed
vertices and improve these results by providing more detailed descriptions of
the distributional limits. Our analysis is based on a correspondence between
the random recursive tree and a representation of the Kingman $n$-coalescent.
