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=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.
Highlights
Graph theory and its applications has received increasing attention in recent years [1,2,3,4,5], which has paved the way for more directions of research.In labeled graph enumeration problems, the vertices of the graph are labeled to be distinguishable from each other, while in unlabeled graph enumeration problems any admissible permutation of the vertices is regarded as producing the same graph, so the vertices are considered unlabeled
The orbit decomposition [10] is an important tool for reducing unlabeled problems to labeled ones: Each unlabeled class is considered to be a symmetry class, or an isomorphism class, of labeled graphs
The enumerative polynomial defined by Equation (3) has the following three properties: (I) P(1) = |Ω|, (II) P (1) = α, the total number of nontrivial automorphisms when counted across all elements of Ω, (III) P(x)dx = |Λ|/|Γ|
Summary
Graph theory and its applications (polyhedra, enumeration, coloring, fullerenes, etc.) has received increasing attention in recent years [1,2,3,4,5], which has paved the way for more directions of research. All vertex-labeled cycles of length 5 are isomorphic and represent the same unlabeled graph, C5, up to isomorphism The vertices of this graph can be assigned labels 1, 2, 3, 4, 5 in twelve different ways. Two triangulations with the same vertex-labeled graph are considered different provided one has a face determined by some three vertices with specific labels while the other does not. As the main result of the current paper, it is shown ( Theorem 2, Section 6) how to generate all different triangulations of the torus, totaling 12 in number, with the vertexlabeled graph G = K2,2,2,2 in an intelligent fashion without using computer resources. The importance of the classification obtained is seen in the geometric realization: Geometrically, the 12 vertex-labeled triangulations correspond to different (as point-sets) 2-dimensional toroidal subcomplexes of the 16-cell in R4. As a byproduct, we obtain all two-dimensional tori in the 2-skeleton of the 16-cell; their realization in a Schlegel diagram of the 16-cell would lead to new toroidal polyhedra in R3 (a Schlegel diagram is a projection of the polytope from R4 into R3 through a point just outside one of its facets)
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