Expected Gutman index of Markov-dependent random polyomino chains

Azami, Laila; Jafari Rad, Nader

doi:10.22108/toc.2026.149467.2373

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	Expected Gutman index of Markov-dependent random polyomino chains
Transactions on Combinatorics
مقالات آماده انتشار، اصلاح شده برای چاپ، انتشار آنلاین از تاریخ 07 مهر 1405 اصل مقاله (480.25 K)
نوع مقاله: Research Paper
شناسه دیجیتال (DOI): 10.22108/toc.2026.149467.2373
نویسندگان
Laila Azami؛ Nader Jafari Rad^*
Department of Mathematics, Shahed University, Tehran, Iran
چکیده
We introduce a Markov-dependent random model for polyomino chains in which each new square is attached linearly or as a kink according to a stationary two-state Markov chain. This model retains the geometric construction of classical random polyomino chains while allowing consecutive attachment types to be dependent. We develop a direct recursive method based on weighted distance sums associated with the two exposed edges of the terminal square. This method yields exact recursive relations for the auxiliary distance quantities and leads to a closed formula for the expected Gutman index. The independent Bernoulli model is recovered when the two transition probabilities are equal. The deterministic linear and zigzag polyomino chains are also obtained as limiting cases. Our results show that the expected Gutman index depends not only on the stationary frequency of kink attachments, but also on the dependence between consecutive attachments.
کلیدواژه‌ها
Gutman index؛ random polyomino chain؛ Markov chain؛ expected value؛ topological index

مراجع
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Expected Gutman index of Markov-dependent random polyomino chains