Graph Invariants Of Fullerenes And Their Statistical Analysis

Fullerene is the most important discovery in the chemistry field recent years. Be-cause of it, Kroto, Smally and Curl won the chemistry Nobel prize in1996. Theseatomic spherical clusters of carbon have farreaching impact on various fields, such asphysical chemistry, material science, biology, medical science, astronomy and geology.The basis topic of the research areas in fullerene chemistry is the stability offullerenes, and graph invariants are important predictors. Equivalence relation betweentwo important graph invariants is proved by characterizing molecular graph structuresof fullerenes. Following the conclusions about the permanent of fullerenes that thebound estimation and the relation with the Kekule′count, and utilizing the graph the-ory, we give a new lower bound of fullerenes’ Kekule′count. By efectively using thespecial properties of the fullerene molecule structure, the stability of the fullerene iso-mers is investigated with many kinds of graph invariants. We propose an optimizationmodel to select the invariants who have stronger correlation with the stable isomers,and several invariants are obtained that they can partly account for the stability indeed.In this thesis we focus on several mathematical problems in chemical graph theoryarising from the graph invariants of fullerenes. It has the following innovative points:1. Fowler-Manolopoulos predictor is proved to be equal to a simple structure pa-rameter.2. The upper bound for the number of the Sachs graphs in fullerenes is proposed.It is further connected with the conclusions of fullerenes’ permanent, so that abetter lower bound of fullerenes’ Kekule′count is obtained.3. As a solid foundation of fullerenes research with data, a great number of calcu-lations are carried out for more than twenty graph invariants.4. A comprehensive study of examining the correlation between graph invariantsand the stable isomers is done. An optimization model is designed and solvedby statistics method. It turns out that the stronger links are found between a few graph invariants and the stability of fullerenes. It is suggested that theseinvariants are useful selectors with over90%of more stable isomers in the rangeof C₆₀～C₁₀₀.