{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,3]],"date-time":"2026-03-03T20:03:21Z","timestamp":1772568201447,"version":"3.50.1"},"reference-count":69,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2020,7,27]],"date-time":"2020-07-27T00:00:00Z","timestamp":1595808000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Romanian Ministry of Education and Research","award":["PN-19-22-01-02"],"award-info":[{"award-number":["PN-19-22-01-02"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>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 M\u00f6bius 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 \u2265 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\u2019s theory) with the space-curvature sphericity\/roundness of general relativity, as is also currently advocated by Wolfram\u2019s language unification of matter-physical phenomenology.<\/jats:p>","DOI":"10.3390\/sym12081233","type":"journal-article","created":{"date-parts":[[2020,7,30]],"date-time":"2020-07-30T12:15:38Z","timestamp":1596111338000},"page":"1233","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":10,"title":["Topological Symmetry Transition between Toroidal and Klein Bottle Graphenic Systems"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9781-1541","authenticated-orcid":false,"given":"Mihai V.","family":"Putz","sequence":"first","affiliation":[{"name":"Laboratory of Structural and Computational Physical-Chemistry for Nanosciences and QSAR, Biology-Chemistry Department, Faculty of Chemistry, Biology, Geography, West University of Timisoara, Str. Pestalozzi No. 16, 300115 Timisoara, Romania"},{"name":"Laboratory of Renewable Energies-Photovoltaics, R&amp;D National Institute for Electrochemistry and Condensed Matter, Dr. A. Paunescu Podeanu Str. No. 144, RO-300569 Timisoara, Romania"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0842-2684","authenticated-orcid":false,"given":"Ottorino","family":"Ori","sequence":"additional","affiliation":[{"name":"Actinium Chemical Research Institute, Via Casilina 1626\/A, 00133 Rome, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,7,27]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Weeks, J.R. (2001). The Shape of Space, CRC Press.","DOI":"10.1201\/9780203912669"},{"key":"ref_2","unstructured":"Weisstein, E.W. (2020, May 08). \u201cKlein Bottle\u201d. From MathWorld\u2014A Wolfram Web Resource. 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