The vertex $k$-partiteness $v_k(G)$ of graph $G$ is defined as the fewestnumber of vertices whose deletion from $G$ yields a $k$-partite graph. In thispaper, we introduce two concepts: monotonic decreasing topological index andmonotonic increasing topological index, and characterize the extremal graphshaving the minimum Wiener index, the maximum Harry index, the maximumreciprocal degree distance, the minimum eccentricity distance sum, the minimumadjacent eccentric distance sum index, the maximum connective eccentricityindex, the maximum Zagreb indices among graphs with a fixed number $n$ ofvertices and fixed vertex $k$-partiteness, respectively.
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机译：图$ G $的顶点$ k $-局部性$ v_k（G）$定义为从$ G $中删除后产生$ k $-局部图的最少数量的顶点。在本文中，我们引入了两个概念：单调递减拓扑指数和单调递增拓扑指数，并描述了具有最小维纳指数，最大哈里指数，最大倒数距离，最小偏心距离总和，最小相邻偏心距离总和指数的极值图，最大连接偏心率指数，最大顶点萨格勒布指数之间具有固定数目的顶点和固定的顶点$ k-部分的图。
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