On The Spectral Radii Of Bicyclic Graphs With Fixed Matching Number

The spectral radius of the connect graph has been studied in depth.In this paper,we give the first ten spectral radii of bicyclic graphs when n≥12 .There are five sections in this dissertation.Section one is introduction.lt introduce the developing about the spectral radius of graphs.In section two,we introduce the background and terminologies.lt contains the concepts of graph, maT_ching,bicyclic and the classes of the bicyclic.We also give the denote of some special graphs and some lemmas respect to the spectral radius of graphs proved by others.Section three mainly study how graft translation for bicyclic graphs,satisfied the spectral radius after graft translation is larger than the spectral radius before graft translation,and the maT_ching number is invariable.Three types of bicyclic graphs are B_n⁺,B_n⁺⁺,θ_n.In these section , we proved all not isomorphism graphs that three kind of bicyclic graphs through satisfies above condition moving.And has drawn below the main conclusion:Theorem3.3 If the connect graph G∈B_n⁺ or B_n⁺⁺, and the maT_ching number of graph G isμ, thenρ(G)＜ρ(G^*),G^* is graph U_3-3(s₁,t₁;s₂,t₂;s₃,t₃;s₄,t₄;s₅,t₅), there s_i,t_i are all nonnegative integer (i = 1, 2, 3, 4,5), and the maT_ching number of G^* isμ.Theorem3.4 If the connect graph G∈θ_n, and the maT_ching number of graph G isμ, thenρ(G)＜ρ(G^*),G^* is graph U_2-1-2(s₁,t₁;s₂,t₂;s₃,t₃;s₄,t₄), there s_i,t_i are all nonnegative integer (i = 1,2,3,4), and the maT_ching number of G^* isμ.Section four mainly studied ifμ≥3, the three types of the bicyclic graph B_n⁺,B_n⁺⁺,θ_nhas the relation of spectral radius after inoculation distortion,and find the first five spectral radii of the bicyclic graphs B_n⁺,B_n⁺⁺,θ_n . There we obtain the relation of spectral radii by calculation the characteristic polynomial of the graphs,and compare them.And has drawn below the main conclusion:Theorem4.3 If graph G∈B_n⁺ or B_n⁺⁺,n≥15 and the maT_ching number isμ≥3,then(1)ρ(G)≤ρ(U_3-3(0,0;0,0;n-2μ+1,μ-3)),with equality if and only if G≌U_3-3(0,0;0,0;n-2μ+1,μ-3).(2)If G(?)U_3-3(0,0;0,0;n-2μ+1,μ-3),thenρ(G)≤ρ(U_3-3(1,0;1,0;n-2μ+1,μ-4)),with equality if and only if G≌U_3-3(1,0; 1,0;n-2μ+1,μ-4).(3)If G≌U_3-3(1,0;1,0;n-2μ+1,μ-4),U_3-3(1,0;1,0;n-2μ+1,μ-4),thenρ(G)≤ρ(U_3-3(1,0;0,0;n-2μ,μ-3)),with equality if and only if G≌U_3-3(1,0;0,0;n-2μ,μ-3).(4)If G(?)U_3-3(1,0;1,0;n-2μ+1,μ-4),U_3-3(1,0;1,0;n-2μ+1,μ-4),U_3-3(1,0;0,0;n-2μ,μ-3), thenρ(G)≤ρ(U'_3-3(0,0;0,0;n-2μ,μ-3)),with equality if and onlyif G≌U'_3-3(0,0;0,0;n-2μ,μ-3)). (5)If G(?)U_3-3(1,0;1,0;n-2μ+1,μ-4),U_3-3(1,0;1,0;n-2μ+1,μ-4),U_3-3(1,0;0,0;n-2μ,μ-3),U'_3-3(0,0;0,0;n-2μ,μ-3), thenρ(G)≤ρ(U_3-3(0,1;0,0;n-2μ,μ-4)),with equality if and only if G≌U_3-3(0,1;0,0;n-2μ,μ-4).Theorem4.6 If graph G∈θ_n,n≥12 and the matching number isμ≥3,then(1)ρ(G)≤ρ(U_2-1-2(1,0;0,0;0,0;n-2μ+1,μ-3)),with equality if and only if G≌U_2-1-2(1,0;0,0;0,0;n-2μ,1,μ-3).(2)If G(?)U_2-1-2(1,0;0,0;0,0;n-2μ+1,μ-3),thenρ(G)≤ρ(U_2-1-2(0,0;0,0;0,0;n-2μ,μ-2)),with equality if and only if G≌U_2-1-2(0,0;0,0;0,0;n-2μ,μ-2).(3) If G(?)U_2-1-2(1,0;0,0;0,0;n-2μ,1,μ-3),U_2-1-2(0,0;0,0;0,0;n-2μ,μ-2), thenρ(G)≤ρ(U_2-1-2(1,0;1,0;1,0;n-2μ+1,μ-4)),with equality if and only if G≌U_2-1-2(1,0;1,0;1,0;n-2μ,1,μ-4).(4) If G(?)U_2-1-2(1,0;0,0;0,0;n-2μ,1,μ-3),U_2-1-2(0,0;0,0;0,0;n-2μ,μ-2),U_2-1-2(1,0;1,0;1,0;n-2μ,1,μ-4), thenρ(G)≤ρ(U_2-1-2(1,0;1,0;0,0;n-2μ,μ-3)),with equality if and only if G≌U_2-1-2(1,0;1,0;0,0;n-2μ,μ-3).(5) If G(?)U_2-1-2(1,0;0,0;0,0;n-2μ,1,μ-3),U_2-1-2(0,0;0,0;0,0;n-2μ,μ-2),U_2-1-2(1,0;1,0;1,0;n-2μ+1,μ-4),U_2-1-2(1,0;1,0;0,0;n-2μ,μ-3),thenρ(G)≤max{ρ(U_2-1-2(1,0;0,0;1,0;n-2μ,μ-3)),ρ(U_2-1-2(0,0;1,0;0,0;n-2μ-1,μ-2))} ,with equality if and only if G≌U_2-1-2(1,0;0,0;1,0;n-2μ,μ-3) or G≌U_2-1-2(0,0;1,0;0,0;n-2μ-1,μ-2).Section five mainly studied the first ten bicyclic graphs with larger spectral radii . First,this section studied whenμ= 2,there are the finite graph class,compare with the spectralradius,from the section four we determine the first ten graphs of the bicyclic graphs.And has drawn below the main conclusion:Theorem5.2 For n≥12 , first ten bicyclic graphs with larger spectral radii are as follows:U_2-1-2(0,0;n-4,0),U_3-3(0,0;n-5,0),U_2-1-2(0,0;1,0;0,0;n-5,0),U_2-1-2(1,0;n-5,0),U_2-1-2(0,0;n-6,1),U_3-3(1,0;0,0;n-6,0),U_3-3(0,0;0,0;n-7,1),U_2-1-2(n-4,0),U_2-1-2(1,0;1,0;0,0;n-6,0),U_2-1-2(0,0;1,0;0,0;n-7,1).