  The vertex $k$-partiteness $v_k(G)$ of graph $G$ is defined as the fewest
number of vertices whose deletion from $G$ yields a $k$-partite graph. In this
paper, we introduce two concepts: monotonic decreasing topological index and
monotonic increasing topological index, and characterize the extremal graphs
having the minimum Wiener index, the maximum Harry index, the maximum
reciprocal degree distance, the minimum eccentricity distance sum, the minimum
adjacent eccentric distance sum index, the maximum connective eccentricity
index, the maximum Zagreb indices among graphs with a fixed number $n$ of
vertices and fixed vertex $k$-partiteness, respectively.
