Topological Properties and Face Index Behavior of TUC4C8 Nanostructures

Abstract

This paper presents a comprehensive analysis of the face index in a class of molecular graphs
derived from the T UC4C8 lattice structure. We investigate the base structure T [m, n], its
subdivision graph S(T ), and the line graph of the subdivision LS(T ), along with their
nanotube and nanotorus variants. Closed-form expressions for the face index are established
for each structure using combinatorial arguments and degree-based face contributions. A
comparative study is conducted by fixing m = 3 and varying n from 2 to 12, with results
compiled in a tabular format. Graphical analysis through 3D surface plots is also provided
to visually demonstrate the trends. The study reveals that subdivision and line graph
operations significantly increase the face index, and torus structures consistently exhibit
higher values than their tube counterparts. These findings highlight the sensitivity of the
face index to topological transformations and its utility in characterizing the complexity of
nanostructured molecular graphs.