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In this manuscript, we delve into the exploration of the first and second Zagreb connection indices of both polyomino chains and random polyomino chains. Our methodology relies on the utilization of Markov chain theory. Within this framework, the article thoroughly examines precise formulas and investigates extreme values. Leveraging the derived formulas, we further explore and elucidate the long-term behavior exhibited by random polyomino chains.
Sigarreta et al. (Mon,) studied this question.