  Recent work of Lazarovich provides necessary and sufficient conditions on a
graph L for there to exist a unique simply-connected (k,L)-complex. The two
conditions are symmetry properties of the graph, namely star-transitivity and
st(edge)-transitivity. In this paper we investigate star-transitive and
st(edge)-transitive graphs by studying the structure of the vertex and edge
stabilisers of such graphs. We also provide new examples of graphs that are
both star-transitive and st(edge)-transitive.
