| Graph theory has derived some branches in a long period of development,which have been applied to physics,chemistry,computer science,communication network and some other fields in the modern world,promoting the development of modern science.The spectral theory of graphs is a very important and active subject in graph theory.In this paper,we mainly discuss the spectral radius of some bicyclic graphs and Halin graphs.First,by subdivision and characteristic polynomial,we get the upper bounds of of two kinds of basic bicyclic graphs are both 2.5616;Additionally,we revealed the relationship between the spectral radius of the graphs obtained by changing the position of pendent edge of a kind of bicyclic graph,through comparing and sorting spectral radius using characteristic polynomial;To verify the above conclusion,we apply Matlab to some specific examples.Then,on the basic of some results,we furthermore discuss the spectral radius of Halin graphs with 2,4,5 interior vertices,respectively.By the graft transformation and the theory of eigenvalue of matrices,we study the increasing of the spectral radius of Halin graphs and some extremal graphs of Halin graphs;We obtained the deduce third largest extremal graph of spectral radius of Halin graphs with 2 interior vertices,get the monotone increasing relations between the Halin graphs with 4 and 5 interior vertices,and discuss the method to estimate upper bond of spectral radius of Halin graphs with 5 interior vertices.In the end,we calculate the spectral radius of some given graphs by Matlab to verify our conclusion.In generally,we mainly use the graft transformation and the properties of characteristic polynomial to study the the spectral radius of some bicyclic graphs and Halin graphs in this paper.Our results will rich the theory of spectral radius in graph theory,and provide theoretical basis for a lot of applications. |
