  Counting the number of maximal independent sets of graphs was started over
$50$ years ago by Erd\H{o}s and Mooser. The problem has been continuously
studied with a number of variations. Interestingly, when the maximal condition
of an independent set is removed, such the concept presents one of topological
indices in molecular graphs, the so called Merrifield-Simmons index. In this
paper, we applied the concept of bivariate generating function to establish the
recurrence relations of the numbers of maximal independent sets of regualr
$n$-gonal cacti when $3 \leq n \leq 6$. By the ideas on meromorphic functions
and the growth of power series coefficients, the asymptotic behaviors through
simple functions of these recurrence relations have been established.
