Isomorphisms of unit distance graphs of layers

For any $\varepsilon \in (0,+\infty)$, consider the metric spaces $\mathbb{R} \times [0,\varepsilon]$ in the Euclidean plane named layers or strips. B. Baslaugh in 1998 found the minimal width $\varepsilon \in (0,1)$ of a layer such that its unit distance graph contains a cycle of a given odd length $k$. The first of the main results of this paper is the fact that the unit distance graphs of two layers $\mathbb{R} \times [0,\varepsilon_1], \mathbb{R} \times [0,\varepsilon_2]$ are non-isomorphic for any different values $\varepsilon_1,\varepsilon_2 \in (0,+\infty)$. We also get a multidimensional analogue of this theorem. For given $n,m \in \mathbb{N}, p \in (1,+\infty), \varepsilon \in (0,+\infty)$, we say that the metric space on $\mathbb{R}^n \times [0,\varepsilon]^m$ with the metric space distance generated by $l_p$-norm in $\mathbb{R}^{n+m}$ is a layer $L(n,m,p,\varepsilon)$. We show that the unit distance graphs of layers $L(n,m,p,\varepsilon_1), L(n,m,p,\varepsilon_2)$ are non-isomorphic for $\varepsilon_1 \neq \varepsilon_2$. The third main result of this paper is the theorem that, for $n \geq 2, \varepsilon > 0$, any automorphism $\phi$ of the unit distance graph of layer $L = L(n,1,2,\varepsilon) = \mathbb{R}^n \times [0,\varepsilon]$ is an isometry. This is related to the Beckman-Quarles theorem of 1953, which states that any unit-preserving mapping of $\mathbb{R}^n$ is an isometry, and to the rational analogue of this theorem obtained by A. Sokolov in 2023.