Extreme Values Of The First General Zagreb Index In Tricyclic Graphs

Graph theory is a new branch of mathimatics. Molecular topology is a graph theory, chemical, computer and other disciplines combined product of these researches, a central issue is to find some physical or chemical properties of molecules. Molecular topology theory is based on invariables, it is trying to establish a relationship between this topology invariables and elements of the physical and chemical properties of molecules. As a result, topology index has a great value on chemical, medical and other aspects. Wiener made the concept of Wiener index, then molecular topology index developed rapidly. This thesis mainly studies one of these topology indeices——Zagreb index.Zagreb index reflects the level of the branch of the molecular skeleton and energy-related elements. The main elements for the molecular design complexity, and so on. This thesis mainly studies tricyclic graphs with the extremal values of the first general Zagreb index(including the largest, the second and third largest values of the first general Zagreb index, the smallest, the second and third smallest values of this index).The first Chapter firstly introduces some basic concepts of graph, some important topological indeices and the researched results about the first general Zagreb index (mainly about trees, unicyclic graphs,bicyclic graphs), then give a brief overview to the main results of the thesis. The second Chapter characterizes all the tricyclic graphs with the largest, the second and third largest values of the first general Zagreb index, and the tricyclic graphs with the smallest, the second and third smallest values of this index. The third Chapter proposes some problems for further research on Zagreb indices.