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 . Numerous topological indices have been scrutinized across various types of random chains, including random cyclooctane chains , random polyphenyl chains , random phenylene chains , random spiro chains , random hexagonal chains , and random polyomino chains . 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 , representing an intersection of these aforementioned research endeavors.