The investigation of symmetrical $q$-extensions of a $d$-dimensional cubic 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 phisycal theories. Earlier V. Trofimov proved that there are only finitely many symmetrical 2-extensions of $\Lambda^{d}$ for any positive integer $d$. This paper is the second and concluding part of our work devoted to the description of all, up to equivalence, realizations of symmetrical 2-extensions of $\Lambda^{2}$ (we show that there are 162 such realizations). In the first part of our work, which was published earlier, we found all, up to equivalence, realizations of symmetrical 2-extensions of $\Lambda^{2}$ such that only the trivial automorphism fixes all blocks of the imprimitivity system (87 realizations). In the present paper, we find the remaining realizations of symmetrical 2-extensions of $\Lambda^{2}$.
Keywords: symmetrical extension of a graph, $d$-dimensional grid.
The paper was received by the Editorial Office on November 12, 2016
Elena Aleksandrovna Konoval’chik, Cand. Sci. (Phys.-Math.), Krasovskii Institute of Mathematics
and Mechanics, Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620990 Russia;
Nosov Magnitogorsk State Technical University, Magnitogorsk, 455000 Russia,
e-mail: asmi@imm.uran.ru .
Kirill Viktorovich Kostousov, Cand. Sci. (Phys.-Math.), Krasovskii Institute of Mathematics and
Mechanics, Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620990 Russia,
e-mail: asmi@imm.uran.ru .
REFERENCES
1. Trofimov V.I. Symmetrical extensions of graphs and some other topics in graph theory related with group theory. Proc. Steklov Inst. Math. (Suppl.). 2012, 279, suppl. 1, pp. 107-112. doi: 10.1134/S0081543812090088.
2. Neganova E.A., Trofimov V.I. Symmetrical extensions of graphs. Izv. Math. 2014, vol. 78, no. 4, pp. 809-835. doi: 10.1070/IM2014v078n04ABEH002707.
3. Trofimov V.I. The finiteness of the number of symmetrical 2-extensions of the $d$-dimensional lattice and similar graphs. Proc. Steklov Inst. Math. (Suppl.). 2014, 285, suppl. 1, pp. 169-182. doi: 10.1134/S0081543814050198.
4. Trofimov V.I. Some remarks on symmetrical extensions of graphs. Proc. Steklov Inst. Math. (Suppl.). 2015, 289, suppl. 1, pp. 199-208. doi: 10.1134/S0081543815050181.
5. Konovalchik E.A., Kostousov K.V. Symmetrical 2-extensions of a 2-dimensional grid. I. Trudy Inst. Mat. Mekh. UrO RAN. 2016, vol. 22, no. 1, pp. 159-179 (in Russian).
6. GAP - Groups, Algorithms, Programming - a System for Computational Discrete Algebra. Ver. 4.5.7: [e-resource]. 2012. Available at: http://www.gap-system.org.
7. Bettina Eick, Franz Gahler, Werner Nickel. GAP package Cryst - Computing with crystallographic groupsCryst, Ver. 4.1, e-resource, 2013. Available at: https://www.gap-system.org/Packages/cryst.html.
8. Bettina Eick, Max Horn, Werner Nickel. GAP package Polycyclic. Ver. 2.11, [e-resource, 2013. Available at: https://www.gap-system.org/Packages/polycyclic.html.
