Enumeration of spanning trees in simplicial complexes

The Kirchoff Matrix Tree Theorem states that the number of spanning trees in a graph G is equal to the absolute value of any cofactor of the Laplacian matrix of G. As the theory of simplicial complexes is a generalization of the theory of graphs one would suspect that there is a generalization of the notion of spanning trees to simplicial complexes, such that the number of spanning trees in a given simplicial complex ∆ is counted in a similiar way. We provide such a generalization and show that these trees, weighted with the number of elements of the torsion subgroup of their individual kth homological group, can be counted by the absolute value of the determinant of a certain submatrix of the Laplacian matrix of ∆. This paper also gives an overview of the field of enumeration of trees using the Laplacian matrix.