In the current study, distance-based topological invariants, namely the Wiener number and the topological roundness index, were computed for graphenic tori and Klein bottles (named toroidal and Klein bottle fullerenes or polyhexes in the pre-graphene literature) described as closed graphs with N vertices and 3N/2 edges, with N depending on the variable length of the cylindrical edge LC of these nano-structures, which have a constant length LM of the Mobius zigzag edge. The presented results show that Klein bottle cubic graphs are topologically indistinguishable from toroidal lattices with the same size (N, LC, LM) over a certain threshold size LC. Both nano-structures share the same values of the topological indices that measure graph compactness and roundness, two key topological properties that largely influence lattice stability. Moreover, this newly conjectured topological similarity between the two kinds of graphs transfers the translation invariance typical of the graphenic tori to the Klein bottle polyhexes with size LC ≥ LC, making these graphs vertex transitive. This means that a traveler jumping on the nodes of these Klein bottle fullerenes is no longer able to distinguish among them by only measuring the chemical distances. This size-induced symmetry transition for Klein bottle cubic graphs represents a relevant topological effect influencing the electronic properties and the theoretical chemical stability of these two families of graphenic nano-systems. The present finding, nonetheless, provides an original argument, with potential future applications, that physical unification theory is possible, starting surprisingly from the nano-chemical topological graphenic space; thus, speculative hypotheses may be drawn, particularly relating to the computational topological unification (that is, complexification) of the quantum many-worlds picture (according to Everett’s theory) with the space-curvature sphericity/roundness of general relativity, as is also currently advocated by Wolfram’s language unification of matter-physical phenomenology.

中文翻译：

---

环形和克莱因瓶石墨系统之间的拓扑对称转换

在当前的研究中，计算了基于距离的拓扑不变量，即维纳数和拓扑圆度指数，用于描述为封闭图的石墨环和克莱因瓶（在石墨烯之前的文献中称为环形和克莱因瓶富勒烯或多六边形） N个顶点和3N/2条边，其中N取决于这些纳米结构的圆柱边LC的可变长度，其具有恒定的莫比乌斯之字形边长度LM。所呈现的结果表明，在某个阈值尺寸 LC 上，克莱因瓶立方图在拓扑上与具有相同尺寸 (N, LC, LM) 的环形晶格无法区分。两种纳米结构共享测量图形紧凑性和圆度的拓扑指数的相同值，这是在很大程度上影响晶格稳定性的两个关键拓扑特性。而且，这种新推测的两种图之间的拓扑相似性将石墨烯环面的典型平移不变性转移到大小为 LC ≥ LC 的克莱因瓶多六边形，使这些图具有顶点传递性。这意味着在这些克莱因瓶富勒烯的节点上跳跃的旅行者不再能够仅通过测量化学距离来区分它们。克莱因瓶立方图的这种由尺寸引起的对称转变代表了影响这两个石墨烯纳米系统家族的电子特性和理论化学稳定性的相关拓扑效应。尽管如此，目前的发现提供了一个具有潜在未来应用的原始论点，即物理统一理论是可能的，令人惊讶的是从纳米化学拓扑石墨空间开始。