The subtree number index of a graph, defined as the number of subtrees, attracts much attention recently. Finding a proper algorithm to compute this index is an important but difficult problem for a general graph. Even for unicyclic and bicyclic graphs, it is not completely trivial, though it can be figured out by try and error. However, it is complicated for tricyclic graphs. This paper proposes path contraction carrying weights (PCCWs) algorithms to compute the subtree number index for the nontrivial case of bicyclic graphs and all 15 cases of tricyclic graphs, based on three techniques: PCCWs, generating function and structural decomposition. Our approach provides a foundation and useful methods to compute subtree number index for graphs with more complicated cycle structures and can be applied to investigate the novel structural property of some important nanomaterials such as the pentagonal carbon nanocone.

中文翻译：

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基于路径收缩携带权的三环图子树枚举算法

该树数指数图的定义（定义为子树的数量）最近引起了很多关注。对于普通图形来说，找到合适的算法来计算该索引是一个重要但困难的问题。即使对于单环图和双环图，它也并非完全无关紧要，尽管可以通过尝试和错误来解决。但是，对于三环图来说很复杂。本文基于PCCW，生成函数和结构分解这三种技术，提出了路径收缩携带权重（PCCW）算法，以计算非平凡的双环图和所有15种三环图的子树数索引。