Counting of [network] nodes

The number of nodes reached by going out to network distance r (with r > 1) from any node in the networks on page 477 is (a) 4r - 4, (b) 3r^{2}/2 - 3r/2 + 1, and (c)

First[Select[4r^{3}/9 + 2r^{2}/3 + {2, 5/3, 5/3} r - {10/9, 1, -4/9}, IntegerQ]]

In any trivalent network, the quantity f[r] obtained by adding up the numbers of nodes reached by going distance r from each node must satisfy f[0] = n and f[1] = 3n, where n is the total number of nodes in the network. In addition, the limit of f[r] for large r must be n^{2}. The values of f[r] for all other r will depend on the pattern of connections in the network.