  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.
