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 ) and ρ ( G ) denote the maximum degree and the spectral radius of a graph G , respectively. Let B ( n ) be the set of bicyclic graphs on n vertices, and B ( n , Δ ) = { G ∈ B ( n ) ∣ Δ ( G ) = Δ } . When Δ ≥ ( n + 3 ) / 2 we characterize the graph which alone maximizes the spectral radius among all the graphs in B ( n , Δ ) . It is also proved that for two graphs G 1 and G 2 in B ( n ) , if Δ ( G 1 ) > Δ ( G 2 ) and Δ ( G 1 ) ≥ ⌈ 7 n / 9 ⌉ + 9 , then ρ ( G 1 ) > ρ ( G 2 ) .