Study Of The Spectral Radius Of Tree Graphs And Its Extended Graphs

Graph theory is a very important branch in combinatorics,which has made great contributions to quantum information,quantum computation,quantum chemistry,military command,transportation management and a lot of other modern science domain.In this thesis,the problems of the spectral radius of tree graph and Halin graphs have been mainly studied.In this topic,firstly,Yuan Hong studied the relations between the graphs and the eigenvalues of many different stylish graphs,and obtained the upper(lower)bound of the spectral radius.Jinsong Yuan,Jinlong Shu and chaoquan Zhang et al.gave the sorting of spectral radius of tree graphs and Halin graphs.First,the extremal graph and its upper and lower bound of the 4-treble-starlike trees with maximum degree 4 has been studied.According to different cases,I obtained the maximal and minimum extremal graphs of 4-treble-starlike trees by graft transformation method,and subdivision method on the internal and external paths.Then the upper bound and the lower bound of the spectral radius of 4-treble-starlike trees with maximum degree 4 was obtained respectively,by research on the relations between the degree and second degree of vertices,and some method in algebraic graph theory,which was verified by some specific examples.Secondly,I further studied on the adjacent radius of Halin graphs.There are some results about the radius of Halin graphs.Based on the known results,in this thesis,I applied the graft transformation technique to the Halin graphs with 2 interior vertices and obtained the second largest extremal graph of spectral radius.Then,I obtained that the spectral radius of Halin graphs with 3 interior vertices is monotone increasing by some algebraic graph theory.The new type Halin graphs were obtained by using the graft transformation,and compared the spectral radius to Halin graphs with 3 interior vertices.Finally,I obtained the relations of the spectral radius between different with 3 interior vertices by some method of algebraic graph theory,which was verified by some examples.In this thesis,I mainly studied the spectral radius,extremal graphs of tree graph and Halin graph,and obtained some good result.These results will enrich the theory of tree graph and Halin graph.