  In this article, we investigate the Zagreb index, a kind of graph-based
topological index, of several random networks, including a class of networks
extended from random recursive trees, plain-oriented recursive trees, and
random caterpillars growing in a preferential attachment manner. We calculate
the mean and variance of the Zagreb index for each class. In addition, we prove
that the asymptotic distribution of the Zagreb index for the first class is
normal, and that the asymptotic distribution of the Zagreb index for the second
class is skewed to the right.
