Some combinatorial properties of flag simplicial pseudomanifolds and   spheres
  A simplicial complex $\Delta$ is called flag if all minimal nonfaces of $\Delta$ have at most two elements. The following are proved: First, if $\Delta$ is a flag simplicial pseudomanifold of dimension $d-1$, then the graph of $\Delta$ (i) is $(2d-2)$-vertex-connected and (ii) has a subgraph which is a subdivision of the graph of the $d$-dimensional cross-polytope. Second, the $h$-vector of a flag simplicial homology sphere $\Delta$ of dimension $d-1$ is minimized when $\Delta$ is the boundary complex of the $d$-dimensional cross-polytope. 
Christos A. Athanasiadis