A Review of the Importance of Mathematical Models in Investigating the Properties of Nanostructures

This paper highlights how mathematical modeling and optimization have transformed nanostructure research. Many properties of nanostructures can be determined by analyzing their geometric structure. We illustrate three representative case studies. First, fullerenes are treated as cubic planar graphs to define the Clar number (maximizing disjoint hexagons in a perfect matching) via binary integer programming. Second, $ZnS$ nanoclusters are characterized by distance-matrix topological indices (Wiener, hyper-Wiener, connectivity, Szeged), which correlate with five key electronic properties through power-law regressions. Third, the Fries number of fullerenes (counting hexagons fully saturated in Kekul\'{e} matchings) is computed both by BILP and by a novel concave quadratic-penalty reformulation solved by interior point methods. Together, these examples demonstrate the power of rigorous graph theory, combinatorial optimization, and numerical programming to yield quantitative insights, accelerate prediction, and guide experimental design.