Unicyclic Graphs, Bicyclic Graphs And Molecular Graphs Generalized Zagreb Indicators

In this thesis, we mainly study one of the topics in graph theory with applications: the extremal values of unicyclic graphs and unicyclic chemical graphs of the first general Zagreb index;the extremal values of bicyclic graphs of the second Zagreb index;the extremal values of unicyclic graphs of the general Randic index. A graph is called unicyclic if it is connected and contains a unique cycle. A simple connected graph is called bicyclic if it has n vertices and n + 1 edges.In the first chapter, after introducing some, important topological indices and basic terminology and notations, we give a brief overview to the main results of the thesis.In Chapter 2, we characterize all unicyclic graphs with the smallest, the second and third smallest values of the first general Zagreb index, and the unicyclic graphs with the largest, the second and third largest values of this index;and we characterize the necessary conditions on the extremal values of unicyclic chemical graphs of the first general Zagreb index.In Chapter 3, we characterize all molecular graphs with the second smallest and largest values of the first general Zagreb index.In Chapter 4, we characterize bicyclic graphs with the smallest values of the second Zagreb index, and if α ≥ 1, unicyclic graphs with the smallest value of the general Randic index.We conclude the thesis with Chapter 5 by proposing some problems for further research on the extremal values of the first general Zagreb index, of the reformulated Zagreb index, and the extremal values of the unicyclic graphs of the general Randic index.