One of the crucial problems in combinatorics and graph theory is characterizing extremal structures with respect to graph invariants from the family of chemical trees. Cruz et al. (2020) presented a unified approach to identify extremal chemical trees for degree-based graph invariants in terms of graph order. The exponential augmented Zagreb index () is a well-established graph invariant formulated for a graph as where signifies the degree of vertex , and is the edge set. Due to some special counting features of , it was not covered by the aforementioned unified approach. As a result, the exploration of extremal chemical trees for this invariant was posed as an open problem in the same article. The present work focuses on generating a complete solution to this problem. Our findings offer maximal and minimal chemical trees of in terms of the graph order .

中文翻译：

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化学树指数增广萨格勒布指数开放问题的完整解决方案


组合学和图论中的关键问题之一是表征化学树家族的图不变量的极值结构。克鲁兹等人。 （2020）提出了一种统一的方法来识别基于图顺序的基于度的图不变量的极值化学树。指数增广萨格勒布指数 () 是一个完善的图不变量，为图制定，其中 表示顶点 的度数， 是边集。由于 的一些特殊的计数特性，它没有被上述统一方法涵盖。因此，对这个不变量的极值化学树的探索在同一篇文章中被提出作为一个开放问题。目前的工作重点是为这个问题提供一个完整的解决方案。我们的研究结果提供了按图顺序表示的最大和最小化学树。