Given simple graphs $X$ and $Y$ on the same number of vertices, the friends-and-strangers graph $\mathsf{FS}(X, Y)$ has as its vertices all bijections from $V(X)$ to $V(Y)$, where two bijections are adjacent if and only if they differ on two adjacent elements of $V(X)$ with images adjacent in $Y$. We study the diameters of connected components of friends-and-strangers graphs: the diameter of a component of $\mathsf{FS}(X,Y)$ corresponds to the largest number of swaps necessary to go from one configuration in the component to another. We show that any component of $\mathsf{FS}(\mathsf{Path}_n, Y)$ has $O(n^2)$ diameter and that any component of $\mathsf{FS}(\mathsf{Cycle}_n, Y)$ has $O(n^4)$ diameter, improvable to $O(n^3)$ whenever $\mathsf{FS}(\mathsf{Cycle}_n, Y)$ is connected. These results address an open problem posed by Defant and Kravitz. Using an explicit construction, we show that there exist $n$-vertex graphs $X$ and $Y$ such that $\mathsf{FS}(X,Y)$ has a component with $e^{Ω(n)}$ diameter. This answers a question raised by Alon, Defant, and Kravitz in the negative. As a corollary, we observe that for such $X$ and $Y$, the lazy random walk on this component of $\mathsf{FS}(X,Y)$ has $e^{Ω(n)}$ mixing time. This result deviates from related classical theorems regarding rapidly mixing Markov chains and makes progress on another open problem of Alon, Defant, and Kravitz. We conclude with several suggestions for future research.
CayleyPy Growth: Efficient growth computations and hundreds of new conjectures on Cayley graphs (Brief version)
Institut Curie, CNRS UMR168, Paris, France（居里研究所，法国国家科学研究中心UMR168，法国巴黎）；University of Washington（华盛顿大学）；Three Gorges Mathematical Research Center, China Three Gorges University（三峡数学研究中心，三峡大学）；Sobolev Institute of Mathematics（索博列夫数学研究所）；Novosibirsk State University（新西伯利亚国立大学）；Independent Researcher（独立研究员）；Moscow State University（莫斯科国立大学）；Kazakh-British Technical University（哈萨克-英国技术大学）；Neapolis University Pafos, Cyprus（尼阿波利斯大学帕福斯校区，塞浦路斯）；City St George's, University of London（伦敦大学城市圣乔治学院）；Hebrew University of Jerusalem（耶路撒冷希伯来大学）；ITMO University（ITMO大学）；Moscow Center for Advanced Studies（莫斯科高等研究中心）；National Research University Higher School of Economics（国立研究大学高等经济学院）；Ariel University（阿里埃勒大学）；Saint-Petersburg Pasteur Institute（圣彼得堡巴斯德研究所）；Criteo AI Lab, Berlin, Germany（Criteo人工智能实验室，德国柏林）；University of Bologna（博洛尼亚大学）；Faculty of Bioengineering and Bioinformatics, Lomonosov Moscow State University（莫斯科国立大学生物工程与生物信息学系）；Researcher at Banner Stat（Banner Stat研究员）；MIREA — Russian Technological University（MIREA — 俄罗斯技术大学）；Research Center of the Artificial Intelligence Institute, Innopolis University（因诺波利斯大学人工智能研究所研究中心）；Inpdependent Researcher（独立研究员）；LIX, Ecole Polytechnique/CNRS, IP Paris（LIX，巴黎综合理工学院/法国国家科学研究中心，巴黎理工学院）
