  Read-once Oblivious Algebraic Branching Programs (ROABPs) compute polynomials
as products of univariate polynomials that have matrices as coefficients. In an
attempt to understand the landscape of algebraic complexity classes surrounding
ROABPs, we study classes of ROABPs based on the algebraic structure of these
coefficient matrices. We study connections between polynomials computed by
these structured variants of ROABPs and other well-known classes of polynomials
(such as depth-three powering circuits, tensor-rank and Waring rank of
polynomials).
  Our main result concerns commutative ROABPs, where all coefficient matrices
commute with each other, and diagonal ROABPs, where all the coefficient
matrices are just diagonal matrices. In particular, we show a somewhat
surprising connection between these models and the model of depth-three
powering circuits that is related to the Waring rank of polynomials. We show
that if the dimension of partial derivatives captures Waring rank up to
polynomial factors, then the model of diagonal ROABPs efficiently simulates the
seemingly more expressive model of commutative ROABPs. Further, a commutative
ROABP that cannot be efficiently simulated by a diagonal ROABP will give an
explicit polynomial that gives a super-polynomial separation between dimension
of partial derivatives and Waring rank.
  Our proof of the above result builds on the results of Marinari, M\"oller and
Mora (1993), and M\"oller and Stetter (1995), that characterise rings of
commuting matrices in terms of polynomials that have small dimension of partial
derivatives. The algebraic structure of the coefficient matrices of these
ROABPs plays a crucial role in our proofs.
