Abstract A k -cone c -cyclic graph is the join of the complete graph K k and a c -cyclic graph (if k = 0 , we get the usual connected graph). A k -apex tree (resp., k -apex unicyclic graph) is defined as a connected graph G with a k -subset V k ⊆ V ( G ) such that G − V k is a tree (resp., unicylic graph), but G − X is not a tree (resp., unicylic graph) for any X ⊆ V ( G ) with | X | k . In this paper, we extend those extremal results and majorization theorems concerning connected graphs of Liu et al. (2019) to k -cone c -cyclic graphs. We also use a unified method to characterize the extremal maximum and minimum results of many topological indices in the class of k -apex trees and k -apex unicyclic graphs, respectively. The later results extend the main results of Javaid et al. (2019); Liu et al. (2020) and partially answer the open problem of Javaid et al. (2019). Except for the new majorization theorem, some new techniques are also established to deal with the minimum extremal results of this paper.

中文翻译：

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k-apex 单环图（树）的统一极值结果

摘要 A k -cone c -循环图是完全图 K k 和 ac -循环图的连接（如果 k = 0 ，我们得到通常的连通图）。一个 k-apex 树（相应地，k-apex 单环图）被定义为一个连通图 G，其具有 ak-子集 V k ⊆ V ( G ) 使得 G − V k 是一棵树（相应地，单环图），但 G − X 不是任何 X ⊆ V ( G ) 的树（或单环图），其中 | X | 。在本文中，我们扩展了 Liu 等人关于连通图的极值结果和专业化定理。(2019) 到 k -cone c -循环图。我们还使用统一的方法来分别刻画 k-apex 树和 k-apex 单环图类中许多拓扑索引的极值最大值和最小值结果。后来的结果扩展了 Javaid 等人的主要结果。(2019); 刘等人。(2020) 并部分回答了 Javaid 等人的开放问题。(2019)。除了新的专业化定理外，还建立了一些新的技术来处理本文的最小极值结果。