Enumeration of Abstract Complexes of a Given Kind (Labeled vs. Unlabeled): A Mathematical Approach

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 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. 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 Q8 with the three imaginary quaternions i,j,k as generators.