Study Of The Problem Of The Spectral Radius Of Cyclic Graphs

Graph theory is an important branch of combination mathematics. It has been applied in many different fields, such as quantum information, quantum computation, quantum chemistry, as well as in computer science and engineering, and the graph theory promoted the development of them.The thesis studies the problem of the spectral radius of unicyclic and bicyclic graphs. In this field, Hong Yuan who firstly studied the relationship between graphs and its eigenvalues, and obtained the upper(lower) bound of the spectral radius of unicyclic graphs with order n.In this thesis, it mainly studies the spectral radius of the cyclic graphs. Firstly, the extremal graphs of the unicyclic graphs with girth g and k pendant vertices are studied, then the upper bound of the spectral radius is calculated, and the spectral radius of this graph is sorted. Extremal graph is given by using the graft translation and matching. Then the relationship of the degree and second degree has been used, the upper bound of the spectral radius of unicyclic graphs can be obtained is 1 +(n -2)~1/2. Secondly, the graft transformation and the skill of eigenvalue computing are used, then the relationship of the cut vertices and spectral radius are discussed. This method is extended to the study of the third spectral radius of unicyclic graphs.Finally, the spectral radius of the bicyclic graphs is studied. The top five largest spectral radius of the bicyclic graphs and the corresponding graphs has been given by the existing literature. In this thesis, the method of contracting and the property of the characteristic polynomial are used, and by the relationship of the position between tree graph and bicyclic graph, it can be discussed in 4 cases. Then the sixth and the seventh largest spectral radius of the bicyclic graphs are obtained, and giving the corresponding of the extremal graphs bicyclic graphs.The problem of the spectral radius of the graph is important in graph theory. The method adopted in this thesis has more extensive adaptability in other graphs.