Sharp upper bounds on the second largest eigenvalues of connected graphs

Let λ2 be the second largest eigenvalue of a graph. Powers (1988) [4] gave some upper bounds of λ2 for general graphs and bipartite graphs, respectively. Considering that these bounds are not always attainable for connected graphs, we present sharp upper bounds of λ2 for connected graphs and connected bipartite graphs in this paper. Moreover, the extremal graphs are completely characterized.