Sharp upper bounds on the second largest eigenvalues of connected graphs

Let ${\lambda }_2$ be the second largest eigenvalue of a graph. Powers (1988) 4] gave some upper bounds of ${\lambda }_2$ for general graphs and bipartite graphs, respectively. Considering that these bounds are not always attainable for connected graphs, we present sharp upper bounds of ${\lambda }_2$ for connected graphs and connected bipartite graphs in this paper. Moreover, the extremal graphs are completely characterized.