  A class of countable infinite graphs with unbounded vertex degree is
considered. In these graphs, the vertices of large degree `repel' each other,
which means that the path distance between two such vertices cannot be smaller
than a certain function of their degrees. Assuming that this function increases
sufficiently fast, we prove that the number of finite connected subgraphs
(animals) of order N containing a given vertex x is exponentially bounded in N
for N belonging to an infinite subset N_x of natural numbers. Under a less
restrictive condition, the same result is obtained for the number of simple
paths originated at a given vertex. These results are then applied to a number
of problems, including estimating the growth of the Randi\'c index and of the
number of greedy animals.
