  We study weight posets of weight multiplicity free (=wmf) representations $R$
of reductive Lie algebras. Specifically, we are interested in relations between
$\dim R$ and the number of edges in the Hasse diagram of the corresponding
weight poset, $# E(R)$. We compute the number of edges and upper covering
polynomials for the weight posets of all wmf-representations. We also point out
non-trivial isomorphisms between weight posets of different irreducible
wmf-representations.
  Our main results concern wmf-representations associated with periodic
gradings or Z-gradings of simple Lie algebras. For Z-gradings, we prove that
$0< 2dim R-# E(R) < h$, where $h$ is the Coxeter number of $\mathfrak g$. For
periodic gradings, we prove that $0\le 2dim R-# E(R)$.
