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Article

Generating the Triangulations of the Torus with the Vertex-Labeled Complete 4-Partite Graph K2,2,2,2

1
Institute of Service Technologies, Russian State University of Tourism and Service, 99 Glavnaya Street, Cherkizovo, Pushkinsky District, 141221 Moscow Region, Russia
2
Department of Discrete Mathematics and Informatics, Dagestan State University, 43-A Gadjieva, 367000 Makhachkala, Russia
*
Author to whom correspondence should be addressed.
Academic Editor: Lorentz Jäntschi
Symmetry 2021, 13(8), 1418; https://doi.org/10.3390/sym13081418
Received: 9 July 2021 / Revised: 28 July 2021 / Accepted: 30 July 2021 / Published: 3 August 2021
(This article belongs to the Special Issue Topological Graph Theory and Discrete Geometry)
Using the orbit decomposition, a new enumerative polynomial P(x) is introduced for abstract (simplicial) complexes of a given type, e.g., trees with a fixed number of vertices or triangulations of the torus with a fixed graph. The polynomial has the following three useful properties. (I) The value P(1) is equal to the total number of unlabeled complexes (of a given type). (II) The value of the derivative P(1) is equal to the total number of nontrivial automorphisms when counted across all unlabeled complexes. (III) The integral of P(x) from 0 to 1 is equal to the total number of vertex-labeled complexes, divided by the order of the acting group. The enumerative polynomial P(x) is demonstrated for trees and then is applied to the triangulations of the torus with the vertex-labeled complete four-partite graph G=K2,2,2,2, in which specific case P(x)=x31. The graph G embeds in the torus as a triangulation, T(G). The automorphism group of G naturally acts on the set of triangulations of the torus with the vertex-labeled graph G. For the first time, by a combination of algebraic and symmetry techniques, all vertex-labeled triangulations of the torus (12 in number) with the graph G are classified intelligently without using computing technology, in a uniform and systematic way. It is helpful to notice that the graph G can be converted to the Cayley graph of the quaternion group Q8 with the three imaginary quaternions i, j, k as generators. View Full-Text
Keywords: group action; orbit decomposition; polynomial; graph; tree; triangulation; torus; automorphism; quaternion group group action; orbit decomposition; polynomial; graph; tree; triangulation; torus; automorphism; quaternion group
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MDPI and ACS Style

Lawrencenko, S.; Magomedov, A.M. Generating the Triangulations of the Torus with the Vertex-Labeled Complete 4-Partite Graph K2,2,2,2. Symmetry 2021, 13, 1418. https://doi.org/10.3390/sym13081418

AMA Style

Lawrencenko S, Magomedov AM. Generating the Triangulations of the Torus with the Vertex-Labeled Complete 4-Partite Graph K2,2,2,2. Symmetry. 2021; 13(8):1418. https://doi.org/10.3390/sym13081418

Chicago/Turabian Style

Lawrencenko, Serge, and Abdulkarim M. Magomedov. 2021. "Generating the Triangulations of the Torus with the Vertex-Labeled Complete 4-Partite Graph K2,2,2,2" Symmetry 13, no. 8: 1418. https://doi.org/10.3390/sym13081418

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