Version 1
: Received: 11 July 2021 / Approved: 12 July 2021 / Online: 12 July 2021 (13:22:09 CEST)
Version 2
: Received: 28 July 2021 / Approved: 29 July 2021 / Online: 29 July 2021 (17:19:45 CEST)

Lawrencenko, S.; Magomedov, A.M. Generating the Triangulations of the Torus with the Vertex-Labeled Complete 4-Partite Graph K_{2,2,2,2}. Symmetry2021, 13, 1418.
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.

Lawrencenko, S.; Magomedov, A.M. Generating the Triangulations of the Torus with the Vertex-Labeled Complete 4-Partite Graph K_{2,2,2,2}. Symmetry2021, 13, 1418.
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.

Abstract

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 = K_{2,2,2,2}, in which specific case P(x) = x^{31}. 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 (twelve 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 Q_8 with the three imaginary quaternions i, j, k as generators.

Keywords

group action; orbit decomposition; polynomial; graph; tree; triangulation; torus; automorphism; quaternion group

Subject

Computer Science and Mathematics, Computer Vision and Graphics

Copyright:
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.