The Extremal Values of Some Monotonic Topological Indices in Graphs with Given Vertex Bipartiteness

Let I(G) be a topological index of a graph G. If I(G + e) < I(G) (or I(G + e) > I(G), respectively) for each edge e is not an element of G, then I(G) decreases (or increases, respectively) with addition of edges. The vertex bipartiteness of a graph is the minimum number of vertices whose deletion from G results in a bipartite graph. In this paper, we determine the extremal values of some monotonic topological indices and characterize the corresponding extremal graphs among all graphs with a given vertex bipartiteness.