Abstract Let G = ( V , E ) be a graph. The inverse sum indeg index (ISI index in short) of G is defined as I S I ( G ) = ∑ v i v j ∈ E d i d j / ( d i + d j ) , where di is the degree of vertex vi. This recently developed topological index can predict total surface area for octane isomers. It is known that the star Sn uniquely minimizes the ISI index among n-vertex trees. However, characterizing n-vertex tree(s) with maximal ISI index (optimal trees, for convenience) appears to be difficult. There are even no sound conjectures on their structure up to now. Let π be a degree sequence, and C ( π ) the set of connected graphs whose degree sequence is π. In the present paper, it is proven that, there exists a so-called BFS-graph that maximizes the ISI index in C ( π ) . This enables us to conduct a computer search for n-vertex optimal tree(s) up to n = 150 . Based on the search outputs, some structural properties of an optimal tree are observed and proven. Finally, some problems and conjectures are proposed, which may indicate the directions for further work towards the complete characterization of optimal trees.

中文翻译：

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在具有最大逆和 indeg 索引的连通图和树上

摘要 令 G = ( V , E ) 为图。G 的逆和 indeg 指数（简称 ISI 指数）定义为 ISI ( G ) = ∑ vivj ∈ E didj / ( di + dj ) ，其中 di 为顶点 vi 的度数。这个最近开发的拓扑指数可以预测辛烷值异构体的总表面积。众所周知，星 Sn 唯一地最小化了 n 顶点树中的 ISI 指数。然而，用最大 ISI 指数（为了方便起见，最优树）表征 n 顶点树似乎很困难。迄今为止，对它们的结构甚至没有合理的猜想。设 π 是一个度数序列，C ( π ) 是度数序列为 π 的连通图的集合。在本文中，证明存在一个所谓的 BFS-graph 可以最大化 C ( ) 中的 ISI 指数。这使我们能够对 n = 150 的 n 顶点最优树进行计算机搜索。基于搜索输出，观察并证明了最优树的一些结构特性。最后，提出了一些问题和猜想，这可能为进一步研究优化树的完整表征指明方向。