Part III - Homogeneous integral table algebras of degree three with no nontrivial linear element

The algebras of the title which also have a faithful element are determined to exact isomorphism. All of the simple homogeneous integral table algebra s of degree three are displayed, and the commutative association schemes in which all the nondiagonal relations have valency three and where some rela tion defines a connected graph on the underlying set are classified up to a lgebraic isomorphism.