On carbon nanocones CNCm[n] and their face index

Over recent decades, there has been remarkable advancement in the field of nanomaterials, particularly with the emergence of carbon nanotubes, boron nanoclusters, boron nanowires, and boron nanotubes, among others. Among these nanomaterials, carbon-based variants are particularly noteworthy due to their exceptional specific geometric and electronic properties, which make them vital in the realm of nanotechnology. The face index of a graph is a significant metric in graph theory, providing essential insights into the structural characteristics of a graph. This index explains aspects of connectivity and planarity, thereby enabling researchers to analyze and comprehend the properties of graphs more effectively. In this study, we present calculations of the recently introduced face index for the graphical structure of carbon nanotubes CNC_m[n] that anticipates to assist researchers and experimentalists in developing experiments/procedures aimed at elucidating the unknown physical and chemical properties of carbon nanocones.