Maximal Independent Sets in Polygonal Cacti

Natawat Klamsakul; Pantaree Thengarnanchai; Mattanaporn Suebtangjai; Pailin Kaewperm; Nuttanon Songsuwan; Pawaton Kaemawichanurat

arXiv · 2202.11294

Maximal Independent Sets in Polygonal Cacti

Natawat Klamsakul·Pantaree Thengarnanchai·Mattanaporn Suebtangjai·Pailin Kaewperm·Nuttanon Songsuwan·Pawaton Kaemawichanurat

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Abstract

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.

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Natawat Klamsakul, Pantaree Thengarnanchai, Mattanaporn Suebtangjai, Pailin Kaewperm, Nuttanon Songsuwan, Pawaton Kaemawichanurat. 2022-02-23. Maximal Independent Sets in Polygonal Cacti. https://arxiv.org/abs/2202.11294

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