On degree–based topological indices of random polyomino chains

Abstract

<abstract><p>In this article, we study the degree-based topological indices in a random polyomino chain. The key purpose of this manuscript is to obtain the asymptotic distribution, expected value and variance for the degree-based topological indices in a random polyomino chain by using a martingale approach. Consequently, we compute the degree-based topological indices in a polyomino chain, hence some known results from the existing literature about polyomino chains are obtained as corollaries. Also, in order to apply the results, we obtain the expected value of several degree-based topological indices such as Sombor, Forgotten, Zagreb, atom-bond-connectivity, Randić and geometric-arithmetic index of a random polyomino chain.</p></abstract>

Zagreb Connection Indices on Polyomino Chains and Random Polyomino Chains

“…This is primarily attributed to the fact that the function g 2 involves four variables, whereas the function g 1 comprises three. In fact, as per the findings presented in [22], the recursive formula for the degree-based topological indices on the P C n is governed by a function of two variables. Consequently, the question arises as to when similar behaviors persist and when distinct patterns emerge for different degree-based topological indices.…”

“…With this initial result in hand, we can derive the expression for the variance by calculating the second moment and applying the recursive relation as follows Proof. The demonstration proceeds similarly, with the only variation being that in this particular case, As mentioned above, a recursive formula for degree-based topological indices over P C n was derived in [22] and the asymptotic behavior and other characteristics for certain types of degree-based topological indices over RP C n , such as the generalized Zagreb index, were also studied. Again, we have a recursive formula; only in this case, it is governed by a function of two variables.…”

“…Again, we have a recursive formula; only in this case, it is governed by a function of two variables. Thus, the same procedure (with slight differences) performed for ZC1 n and ZC2 n works, giving us information about the expected value, variance and asymptotic behavior of any degree-based topological index over RP C n (obviously encompassing those already covered in [22]).…”

“…On the other hand, in the realm of random chain analysis, topological indices have emerged as a dynamic research area over the past two decades [9]. Numerous topological indices have been scrutinized across various types of random chains, including random cyclooctane chains [10,11], random polyphenyl chains [12,13], random phenylene chains [14,15], random spiro chains [16,17,18], random hexagonal chains [19,20,21], and random polyomino chains [22,23]. Furthermore, it is noteworthy that the Zagreb connection indices have been investigated within the context of random cyclooctatetraene chains, random polyphenyl chains, and triangular chain structures [24,25], representing an intersection of these aforementioned research endeavors.…”

“…Inspired by the aforementioned research and our previous investigation of certain topological indices based on degree over random polyomino chains and polyomino chains [22], the primary objective of this manuscript is to establish explicit formulas for the first and second Zagreb connection indices. Additionally, we aim to analyze their extreme values within the context of polyomino chains.…”