Some results on the spectral radii of bicyclic graphs

A bicyclic graph is a connected graph in which the number of edges equals the number of vertices plus one. Let Δ(G)Δ(G) and ρ(G)ρ(G) denote the maximum degree and the spectral radius of a graph GG, respectively. Let B(n)B(n) be the set of bicyclic graphs on nn vertices, and B(n,Δ)={G∈B(n)∣Δ(G)=Δ}B(n,Δ)={G∈B(n)∣Δ(G)=Δ}. When Δ≥(n+3)/2Δ≥(n+3)/2 we characterize the graph which alone maximizes the spectral radius among all the graphs in B(n,Δ)B(n,Δ). It is also proved that for two graphs G1G1 and G2G2 in B(n)B(n), if Δ(G1)>Δ(G2)Δ(G1)>Δ(G2) and Δ(G1)≥⌈7n/9⌉+9Δ(G1)≥⌈7n/9⌉+9, then ρ(G1)>ρ(G2)ρ(G1)>ρ(G2).