Counting maximal independent sets in some n-gonal cacti

Counting the number of maximal independent sets of graphs was started over 50 years ago by Erdős 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 regular n-gonal cacti when 3≤n≤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.