  A graph G = (V,E) is called fully regular if for every independent set
$I\subset V$ , the number of vertices in $V\setminus$ I that are not connected
to any element of I depends only on the size of I. A linear ordering of the
vertices of G is called successive if for every i, the first i vertices induce
a connected subgraph of G. We give an explicit formula for the number of
successive vertex orderings of a fully regular graph.
  As an application of our results, we give alternative proofs of two theorems
of Stanley and Gao + Peng, determining the number of linear edge orderings of
complete graphs and complete bipartite graphs, respectively, with the property
that the first i edges induce a connected subgraph. As another application, we
give a simple product formula for the number of linear orderings of the
hyperedges of a complete 3-partite 3-uniform hypergraph such that, for every i,
the first i hyperedges induce a connected subgraph. We found similar formulas
for complete (non-partite) 3-uniform hypergraphs and in another closely related
case, but we managed to verify them only when the number of vertices is small.
