Structure of the Group of Balanced Labelings on Graphs, its Subgroups and Quotient Groups

We discuss functions from edges and vertices of an undirected graph to an Abelian group. Such functions, when the sum of their values along any cycle is zero, are called balanced labelings. The set of balanced labelings forms an Abelian group. We study the structure of this group and the structure of two closely related to it groups: the subgroup of balanced labelings which consists of functions vanishing on vertices and the corresponding factor-group. This work is completely self-contained, except the algorithm for obtaining the 3-edge-connected components of an undirected graph, for which we make appropriate references to the literature.Comment: 22 page

g-elements, finite buildings and higher Cohen-Macaulay connectivity

The main result is a proof that the g-vector of a simplicial complex with a convex ear decomposition is an M-vector. This is a generalization of similar results for matroid complexes. We also show that a finite building has a convex ear decomposition. This leads to connections between higher Cohen-Macaulay connectivity and increasing h-vectors.Comment: To appear in JCT A. 20 page