Two applications of semi-stable graphs

A Michigan graph G on a vertex set V is called semi-stable if for some @u@eV, @C(G"@u) = @C(G)"@u. It can be shown that all regular graphs are semi-stable and this fact is used to show (i) that if @C(G) is doubly transitive then G = K"n or K@?"n, and (ii) that @C(G) can be recovered from @C(G"@u). The second result is extended to the case of stable graphs.