Neighborhood‎ ‎M-Polynomial of‎ ‎Graph‎ ‎Operations: Exploring Nanostructure Applications and Correcting Cycle-Related Graph Results

‎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.