Generating the Triangulations of the Torus with the Vertex-Labeled Graph K_{2,2,2,2}

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 useful properties: (I) P(1) is equal to the number of unlabeled complexes (of a given type), (II) the derivative P'(1) is equal to the number of non-trivial automorphisms over all unlabeled complexes, (III) the integral of P(x) from 0 to 1 is equal to the 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 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 on 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 three quaternions, i, j, k, as generators.