关于具有最小原子键连接指数IV的树的结构性质：解决关于长度为3的悬挂路的一个猜想
On structural properties of trees with minimal atom-bond connectivity index IV: Solving a conjecture about the pendent paths of length three
摘要：
原子-键连接性（ABC）指数是研究最多的基于度的分子结构描述符之一，具有多种化学应用。已知在所有连通图中，树最小化ABC指数。然而，一个完整的表征树与最小的ABC指数仍然是一个悬而未决的问题。到目前为止，已证明的性质之一是，具有最小ABC指数的树可能最多有一条长度为$3$的悬垂路径，并推测如果树的阶大于$1178$，则不可能是这种情况。在这里，我们提供了一个肯定的答案，加强版本的猜想，表明一棵树与最小的ABC指数不能包含一个悬垂路径长度$3$，如果它的顺序是大于$415$。
英文摘要：
The atom-bond connectivity (ABC) index is one of the most investigated degree-based molecular structure descriptors with a variety of chemical applications. It is known that among all connected graphs, the trees minimize the ABC index. However, a full characterization of trees with a minimal ABC index is still an open problem. By now, one of the proved properties is that a tree with a minimal ABC index may have, at most, one pendent path of length $3$, with the conjecture that it cannot be a case if the order of a tree is larger than $1178$. Here, we provide an affirmative answer of a strengthened version of that conjecture, showing that a tree with minimal ABC index cannot contain a pendent path of length $3$ if its order is larger than $415$.
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