Some Special Classes Of Graph Zagreb Index

A crucial question in combination chemistry is searching molecules with some physico-chemical characteristics. In order to obtain the expected molecules, we firstly get the topological indices of the molecules and compute the value, then establish the database containing all of the molecular graphs with the value.The Zagreb indices are mainly used in molecule design, molecule complexity and so on. They describe the molecule skeleton branch, and concerns with the molecular energy. The Zagreb indices and their improvement forms may not only describe the molecule structure, but analyze the correlation with the characteristics. This thesis mainly studies bicyclic graphs and molecular graphs with the extremal values of the first general Zagreb index, and bicyclic graphs with the extremal values of the second Zagreb index.In Chapter 1, we firstly introduce some important topological indices and basic terminologies, then give a brief overview to the main results of the thesis.In Chapter 2, we characterize all the bicyclic graphs with the smallest, the second and third smallest values of the first general Zagreb index, and the bicyclic graphs with the largest, the second and third largest values of this index.In Chapter 3, we gain some necessary conditions for the molecular graphs to attain the second smallest and largest values of the first general Zagreb index.In Chapter 4, we characterize all the bicyclic graphs with the smallest, the second and third smallest values of the second Zagreb index, and the bicyclic graphs with the largest, the second and third largest values of this index.In Chapter 5, we propose some problems for further research on Zagreb indices.