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
