For a connected graph, the first Zagreb eccentricity index $\xi _{1}$ is defined as the sum of the squares of the eccentricities of all vertices, and the second Zagreb eccentricity index $\xi _{2}$ is defined as the sum of the products of the eccentricities of pairs of adjacent vertices. In this paper, we mainly present a different and universal approach to determine the upper bounds respectively on the Zagreb eccentricity indices of trees, unicyclic graphs and bicyclic graphs, and characterize these corresponding extremal graphs, which extend the ordering results of trees, unicyclic graphs and bicyclic graphs in (Du et al. in Croat. Chem. Acta 85:359–362, 2012; Qi et al. in Discrete Appl. Math. 233:166–174, 2017; Li and Zhang in Appl. Math. Comput. 352:180–187, 2019). Specifically, we determine the n-vertex trees with the i-th largest indices $\xi _{1}$ and $\xi _{2}$ for i up to $\lfloor n/2+1 \rfloor $ compared with the first three largest results of $\xi _{1}$ and $\xi _{2}$ in (Du et al. in Croat. Chem. Acta 85:359–362, 2012), the n-vertex unicyclic graphs with respectively the i-th and the j-th largest indices $\xi _{1}$ and $\xi _{2}$ for i up to $\lfloor n/2-1 \rfloor $ and j up to $\lfloor 2n/5+1 \rfloor $ compared with respectively the first two and the first three largest results of $\xi _{1}$ and $\xi _{2}$ in (Qi et al. in Discrete Appl. Math. 233:166–174, 2017), and the n-vertex bicyclic graphs with respectively the i-th and the j-th largest indices $\xi _{1}$ and $\xi _{2}$ for i up to $\lfloor n/2-2\rfloor $ and j up to $\lfloor 2n/15+1\rfloor $ compared with the first two largest results of $\xi _{2}$ in (Li and Zhang in Appl. Math. Comput. 352:180–187, 2019), where $n\ge 6$ . More importantly, we propose two kinds of index functions for the eccentricity-based topological indices, which can yield more general extremal results simultaneously for some classes of indices. As applications, we obtain and extend some ordering results about the average eccentricity of bicyclic graphs, and the eccentric connectivity index of trees, unicyclic graphs and bicyclic graphs.

中文翻译：

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具有大的基于偏心率的拓扑索引的有序图

对于一个连通图，将第一个Zagreb偏心指数$ \ xi _ {1} $定义为所有顶点的偏心率的平方和，将第二个Zagreb偏心指数$ \ xi _ {2} $定义为相邻顶点对的偏心率的乘积之和。在本文中，我们主要提出一种不同的通用方法来分别确定树木，单环图和双环图的Zagreb偏心率指数的上限，并刻画这些对应的极值图，从而扩展了树木，单环图和树的有序结果。 （Du等人在Croat。Chem。Acta 85：359–362，2012; Qi等人在Discrete Appl。Math。233：166–174，2017; Li and Zhang在Appl。Math.Comput。 352：180–187，2019）。特别，和n个顶点i分别具有第i个和第j个最大索引$ \ xi _ {1} $和$ \ xi _ {2} $的n顶点双环图，直到$ \ lfloor n / 2-2 \ rfloor $和j直到$ \ lfloor 2n / 15 + 1 \ rfloor $，与$ \ xi _ {2} $的前两个最大结果相比（应用数学计算352：180–187中的Li和Zhang ，2019），其中$ n \ ge 6 $。更重要的是，我们为基于偏心率的拓扑索引提出了两种索引函数，它们对于某些类别的索引可以同时产生更一般的极值结果。作为应用，我们获得并扩展了有关双环图的平均偏心率，树，单环图和双环图的偏心连接指数的排序结果。我们为基于偏心率的拓扑索引提出了两种索引函数，它们对于某些类别的索引可以同时产生更一般的极值结果。作为应用，我们获得并扩展了有关双环图的平均偏心率，树，单环图和双环图的偏心连接指数的排序结果。我们为基于偏心率的拓扑索引提出了两种索引函数，它们对于某些类别的索引可以同时产生更一般的极值结果。作为应用，我们获得并扩展了有关双环图的平均偏心率，树，单环图和双环图的偏心连接指数的排序结果。