Some coinductive graphs

Summary LetT be a universal theory of graphs such that Mod(T) is closed under disjoint unions. Let_T be a disjoint union_i such that each_i is a finite model ofT and every finite isomorphism type in Mod(T) is represented in{_ii<3}. We investigate under what conditions onT, Th(_T) is a coinductive theory, where a theory is called coinductive if it can be axiomatizated by -sentences. We also characterize coinductive graphs which have quantifier-free rank 1.