University of Kashan Iranian Journal of Mathematical Chemistry 2228-6489 15 3 2024 09 01 Neighborhood‎ ‎M-Polynomial of‎ ‎Graph‎ ‎Operations: Exploring Nanostructure Applications and Correcting Cycle-Related Graph Results 155 174 114452 10.22052/ijmc.2024.253190.1733 EN Gunajyoti Saharia Department of Mathematics‎, ‎NEHU‎, ‎Shillong‎, ‎India ‎Jibitesh Dutta Mathematics Division‎, ‎Department of Basic Sciences and Social Sciences‎, ‎NEHU‎, ‎Shillong‎, ‎India Sanghita Dutta Department of Mathematics‎, ‎NEHU‎, ‎Shillong‎, ‎India Journal Article 2023 08 04 ‎Mathematical chemistry is a field of mathematics where chemical compounds are studied by associating a graph to it‎. ‎A topological index serves as a mathematical invariant that elucidates the underlying topological arrangement of molecules or networks‎. ‎This paper explores the neighborhood M-Polynomial concerning various graph operations of regular graphs‎. ‎Additionally‎, ‎it addresses and rectifies several erroneous results pertaining to cycle-related graphs that were previously reported‎.‎ Furthermore, we examine the applications of the neighborhood ‎$‎M‎$‎-Polynomial to the ‎$VPHX[m,n]‎‎$‎ nanotubes and $VPHY[m,n]‎$ ‎nanotori, presenting their potential in real-world.‎ Through this comprehensive investigation, we aim to advance the understanding of topological indices and their practical implications. https://ijmc.kashanu.ac.ir/article_114452_3fb7fc7ca420b1af3fa7b23de35f852a.pdf