AI 中文总结
本文构造了纯组合自相似结构模型替换图，给出其为格罗莫夫双曲图的充要条件，还推导了仅依赖生成元G和J的双曲性相关条件。
AI 中文摘要
本文中，我们构造了一类无限图，称为替换图。其顶点集由有限字母表上的所有有限字构成。通过添加连接每个字与其子节点的垂直边，以及由两个有限有向图G和J递归定义的水平边，形成了一个有向图：具有相同父节点的顶点之间的边遵循G，而父节点水平相连的顶点之间的边遵循J。替换图被定义为其基础图。替换图提供了一种纯组合的自相似结构模型，独立于任何底层几何结构。此外，我们建立了替换图为双曲图的充要条件，该条件基于与足够长的最短水平路径相关的路径矩阵的消失来表述。基于这一特征，我们进一步推导了几个仅依赖生成元G和J的双曲性必要或充分条件。
英文摘要
In this paper, we construct a class of infinite graphs, called substitution graphs. The vertex set consists of all finite words over a finite alphabet. A directed graph is formed by adding vertical edges connecting each word to its children and horizontal edges defined recursively by two finite directed graphs G and J: edges among vertices with the same parent follow G, while edges between vertices whose parents are horizontally linked follow J. The substitution graph is defined as its underlying graph. Substitution graphs provide a purely combinatorial model of self-similar structures, independent of any underlying geometric structure. Furthermore, we establish a necessary and sufficient condition for substitution graphs to be hyperbolic, formulated in terms of the vanishing of path matrices associated with sufficiently long shortest horizontal paths. Based on this characterization, we further derive several conditions that are either necessary or sufficient for hyperbolicity, depending only on the generators G and J.
Comments25 pages, 6 figures
