Study On The Spectral Radius Of Bicyclic Graphs

The research on the bicyclic graphs has been for a long time.In 2011,Liu Muhuo and Liu Bolian gave the control theorem of comparing two bicyclic graphs B*for spectral radius under certain conditions,that is,given two of the same order the degree sequence ?,?' of the bicyclic graphs and its corresponding graphs B?*,B?'*,if ?' controls?,then ?(B?*)<?(B?'*).But it is that the other properties of the bicyclic graphs have not been explored,so this article mainly studies the following three aspects' content.1.Study the extremum of the spectral radius by classifying the set Bmax(n)of the bicyclic connected graph B?*,and obtains the bicyclic degree sequence ? corresponding to the extremum;2.Compare the spectral radius of two kinds of bicyclic graphs B?*corresponding to degree sequences without major relation by analyzing the characteristic polynomials of adjacency matrixs of two kinds of bicyclic graphs B?*.3.Study the change of the maximum eigenvalues of two types of connected bicyclic graphs S_n(3,3)and ?_n(3,3)as the change of n;At the same time,it also gives a comparison of the maximum eigenvalues of S_n(3,3)and ?_n(3,3)under the condition of n?5.