Let I(G) be a topological index of a graph. If I(G + e) < I(G) (or I(G + e) > I(G), respectively) for each edge e < G, then I(G) is decreasing (or increasing, respectively) with addition of edges. In this paper, we determine the extremal values of some monotonic topological indices which decrease or increase with addition of edges, and characterize the corresponding extremal graphs among bipartite graphs with a given connectivity