{"title":"Hamiltonian Cycles on Ammann-Beenker Tilings","authors":"Shobhna Singh, Jerome Lloyd, Felix Flicker","doi":"10.1103/physrevx.14.031005","DOIUrl":null,"url":null,"abstract":"We provide a simple algorithm for constructing Hamiltonian graph cycles (visiting every vertex exactly once) on a set of arbitrarily large finite subgraphs of aperiodic two-dimensional Ammann-Beenker (AB) tilings. Using this result, and the discrete scale symmetry of AB tilings, we find exact solutions to a range of other problems which lie in the complexity class NP-complete for general graphs. These include the equal-weight traveling salesperson problem, providing, for example, the most efficient route a scanning tunneling microscope tip could take to image the atoms of physical quasicrystals with AB symmetries; the longest path problem, whose solution demonstrates that collections of flexible molecules of any length can adsorb onto AB quasicrystal surfaces at density one, with possible applications to catalysis; and the three-coloring problem, giving ground states for the <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>q</mi></math>-state Potts model (<math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>q</mi><mo>≥</mo><mn>3</mn></math>) of magnetic interactions defined on the planar dual to AB, which may provide useful models for protein folding.","PeriodicalId":20161,"journal":{"name":"Physical Review X","volume":null,"pages":null},"PeriodicalIF":11.6000,"publicationDate":"2024-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Physical Review X","FirstCategoryId":"101","ListUrlMain":"https://doi.org/10.1103/physrevx.14.031005","RegionNum":1,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"PHYSICS, MULTIDISCIPLINARY","Score":null,"Total":0}
引用次数: 0
Abstract
We provide a simple algorithm for constructing Hamiltonian graph cycles (visiting every vertex exactly once) on a set of arbitrarily large finite subgraphs of aperiodic two-dimensional Ammann-Beenker (AB) tilings. Using this result, and the discrete scale symmetry of AB tilings, we find exact solutions to a range of other problems which lie in the complexity class NP-complete for general graphs. These include the equal-weight traveling salesperson problem, providing, for example, the most efficient route a scanning tunneling microscope tip could take to image the atoms of physical quasicrystals with AB symmetries; the longest path problem, whose solution demonstrates that collections of flexible molecules of any length can adsorb onto AB quasicrystal surfaces at density one, with possible applications to catalysis; and the three-coloring problem, giving ground states for the -state Potts model () of magnetic interactions defined on the planar dual to AB, which may provide useful models for protein folding.
我们提供了一种在一组任意大的有限子图上构建汉密尔顿图循环(每个顶点精确访问一次)的简单算法,这些子图是非周期性的二维安曼-宾克(AB)倾斜图。利用这一结果和 AB 层的离散尺度对称性,我们找到了一系列其他问题的精确解,这些问题的复杂度属于一般图的 NP-完全问题。这些问题包括等权旅行推销员问题,例如,它提供了扫描隧道显微镜尖端对具有 AB 对称性的物理准晶体的原子进行成像的最有效路径;最长路径问题,该问题的求解证明了任何长度的柔性分子集合都能在密度为一的情况下吸附到 AB 类晶体表面,并可能应用于催化作用;以及三着色问题,该问题给出了定义在 AB 平面对偶面上的磁相互作用 q 态波茨模型(q≥3)的基态,这可能为蛋白质折叠提供有用的模型。
期刊介绍:
Physical Review X (PRX) stands as an exclusively online, fully open-access journal, emphasizing innovation, quality, and enduring impact in the scientific content it disseminates. Devoted to showcasing a curated selection of papers from pure, applied, and interdisciplinary physics, PRX aims to feature work with the potential to shape current and future research while leaving a lasting and profound impact in their respective fields. Encompassing the entire spectrum of physics subject areas, PRX places a special focus on groundbreaking interdisciplinary research with broad-reaching influence.