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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 continuouslystudied with a number of variations. Interestingly, when the maximal conditionof an independent set is removed, such the concept presents one of topologicalindices in molecular graphs, the so called Merrifield-Simmons index. In thispaper, we applied the concept of bivariate generating function to establish therecurrence relations of the numbers of maximal independent sets of regualr$n$-gonal cacti when $3 \leq n \leq 6$. By the ideas on meromorphic functionsand the growth of power series coefficients, the asymptotic behaviors throughsimple functions of these recurrence relations have been established.

Maximal Independent Sets in Polygonal Cacti