Geodesic distance, sometimes called shortest path length, has proven useful in a great variety of applications, such as information retrieval on networks including treelike networked models. Here, our goal is to analytically determine the exact solutions to geodesic distances on two different families of growth trees which are recursively created upon an arbitrary tree $\mathcal {T}$ using two types of well-known operations, first-order subdivision and ( $1,m$ )-star-fractal operation. Different from commonly-used methods, for instance, spectral techniques, for addressing such a problem on growth trees using a single edge as seed in the literature, we propose a novel method for deriving closed-form solutions on the presented trees completely. Meanwhile, our technique is more general and convenient to implement compared to those previous methods mainly because there are not complicated calculations needed. In addition, the closed-form expression of mean first-passage time ( $MFPT$ ) for random walk on each member in tree families is also readily obtained according to connection of our obtained results to effective resistance of corresponding electric networks. The results suggest that the two topological operations above are sharply different from each other due to $MFPT$ for random walks, and, however, have likely to show the similar performance, at least, on geodesic distance.

中文翻译：

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一种任意阶树细分的测地距离方法及其应用

测地距离，有时称为最短路径长度，已被证明在多种应用中很有用，例如网络上的信息检索，包括树状网络模型。在这里，我们的目标是分析确定在任意树上递归创建的两个不同生长树族的测地线距离的精确解$\数学{T}$使用两种类型的众所周知的操作，一阶细分和 ( 1美元，百万美元 )-星形分形运算。不同于常用的方法，例如光谱技术，在文献中使用单边作为种子来解决生长树上的此类问题，我们提出了一种新方法，用于在所呈现的树上完全推导封闭形式的解决方案。同时，与以前的方法相比，我们的技术更通用，更方便实现，主要是因为不需要复杂的计算。此外，平均首过时间的封闭式表达式 ( $MFPT$ ) 对于树族中每个成员的随机游走，根据我们获得的结果与相应电网的有效电阻的连接，也很容易获得。结果表明，上述两种拓扑运算彼此截然不同，因为$MFPT$但是，对于随机游走，至少在测地距离上可能会显示出类似的性能。