Symmetrical 2-extensions of the 3-dimensional grid with all connections of type 2
E. A. Konovalchik,
K. V. Kostousov N.N. Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Ekaterinburg
Abstract:
The investigation of symmetrical
$q$-extensions of a
$d$-dimensional grid
$\Lambda^{d}$ is of interest both for group theory and for graph theory. For small
$d\geq 1$ and
$q>1$ (especially for
$q=2$), symmetrical
$q$-extensions of
$\Lambda^{d}$ are of interest for molecular crystallography and some physical theories. Earlier V.I. Trofimov proved that there are only finitely many (up to equivalence) realizations of symmetrical 2-extensions of
$\Lambda^{d}$ for any positive integer
$d$. E.A. Konovalchik and K.V. Kostousov found all, up to equivalence, realizations of symmetrical 2-extensions of the grid
$\Lambda^2$. Then, in the Part I of the study devoted to the case with
$d=3$, K.V. Kostousov found all, up to equivalence, realizations of symmetrical 2-extensions of the grid
$\Lambda^3$ for which only the trivial automorphism preserves all blocks of
$\sigma$. Among other realizations of symmetrical 2-extensions of the grid
$\Lambda^3$ the realizations, in which every vertex is adjacent with only one vertex in each adjacent block, compound an important subclass. In this work we find all of them, up to equivalence.
UDC:
512.54 +
519.17
MSC: 05C63,
20H15 Received: 04.11.2024
Revised: 12.02.2025
Accepted: 18.03.2025
Language: English
DOI:
10.21538/0134-4889-2025-31-4-fon-02
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