Enumeration of Abstract Complexes of a Given Kind (Labeled vs. Unlabeled): A Mathematical Approach| Chapter 7 | Novel Research Aspects in Mathematical and Computer Science Vol. 8

 The utilisation of our enumeration polynomial P(x), a link between the labelled and unlabeled settings, is covered in the current chapter. Using the decomposition of the set of complexes into orbits, we demonstrate how to use the enumeration polynomial P(x) proposed for abstract (simplicial) complexes of a particular form, such as trees with a given number of vertices or torus triangulations with a specified network. Following a demonstration for trees, the enumeration polynomial P(x) is then applied to the triangulations of the torus with the vertex-labeled complete four-partite graph G = K2,2,2,2, where P(x) specifically equals x31. The triangulation T(G) of the graph G embeds it in the torus. In the current chapter we review the use of our enumeration polynomial P(x) which is a bridge between the labeled and unlabeled settings. We show how to apply the enumeration polynomial P(x) introduced for abstract (simplicial) complexes of a specific form, such as trees with a given number of vertices or torus triangulations with a specified network, using the decomposition of the set of complexes into orbits. The enumeration polynomial P(x) is demonstrated for trees and then is applied to the triangulations of the torus with the vertex-labeled complete four-partite graphG=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. There are twelve vertex-labeled triangulations of the torus with the graph G; they are classified intelligently, uniformly, and systematically without the use of computational technology. This is done by using algebraic and symmetry approaches. 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. The set of triangulations of the torus with the vertex-labeled graph G is naturally affected by the automorphism group of G. The torus with the graph G has twelve vertex-labeled triangulations, which are intelligently, uniformly, and systematically classified without the aid of computing technology. Using symmetry and algebraic methods, this is accomplished. It is useful to note that by using the three hypothetical quaternions I j, and k as generators, the graph G can be transformed into the Cayley graph of the quaternion group Q8.

Author(s) Details:

S. Lawrencenko,
Institute of Service Technologies, Russian State University of Tourism and Service, 99 Glavnaya Street, Cherkizovo, Pushkinsky District, Moscow Region, 141221, Russia.

A. M. Magomedov,
Department of Discrete Mathematics and Informatics, Dagestan State University, 43-A Gadjieva, Makhachkala, 367000, Russia.

Please see the link here: https://stm.bookpi.org/NRAMCS-V8/article/view/8361