  We study the statistics of edges and vertices in the vicinity of a reference
vertex (origin) within random planar quadrangulations and Eulerian
triangulations. Exact generating functions are obtained for theses graphs with
fixed numbers of edges and vertices at given geodesic distances from the
origin. Our analysis relies on bijections with labeled trees, in which the
labels encode the information on the geodesic distance from the origin. In the
case of infinitely large graphs, we give in particular explicit formulas for
the probabilities that the origin have given numbers of neighboring edges
and/or vertices, as well as explicit values for the corresponding moments.
