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.