  Consider the tesselation of the hyperbolic plane by m-gons, l per vertex. In
its 1-skeleton, we compute the growth series of vertices, geodesics, tuples of
geodesics with common extremities. We also introduce and enumerate "holly
trees", a family of reduced loops in these graphs. We then apply Grigorchuk's
result relating cogrowth and random walks to obtain lower estimates on the
spectral radius of the Markov operator associated with a symmetric random walk
on these graphs.
