SOME RESULTS ON THE SPECTRAL RADII OF TREES, UNICYCLIC, AND BICYCLIC GRAPHS

Let Delta(G), Delta for short, be the maximum degree of a graph G. In this paper, trees (resp., unicyclic graphs and bicyclic graphs), which attain the first and the second largest spectral radius with respect to the adjacency matrix in the class of trees (resp., unicyclic graphs and bicyclic graphs) with n vertices and the maximum degree Delta, where Delta >= n+1/2 (resp., Delta >= n/2+1 and Delta >= n+3/2) are determined. Moreover, it is shown that the spectral radius of a unicyclic graph U (resp., a bicyclic graph B) on n vertices strictly increases with its maximum degree when Delta(U) >= 1/9(1+ root 6n+10)(2) (resp., Delta(B) >= 1/9(2 + root 6n+28)(2)).