Low molar mass liquid crystal solvents with positive dielectric anisotropy exhibit an electrorheological (ER) effect such that the viscosity, eta(on), in the presence of a strong electric field, applied transverse to the flow, is larger than that, eta(off), in the absence of such a field. Dissolution of a liquid crystal polymer (LCP) enhances the magnitude of the ER effect (eta(on) - eta(off)) by an amount, delta eta(on) - delta eta(off), which is an increasing function of LCP concentration and depends on the molecular architecture of the LCP. Specifically, we show that two main-chain LCPs, with different chemical structures, strongly increase the ER response, a side-on side-chain LCP moderately increases the response, and an end-on side-chain LCP weakly increases the response. These diverse behaviors can be interpreted using theoretical arguments which assume that the LCP conformation is an ellipsoid of revolution whose orientation relative to the flow direction is determined by the balance between the hydrodynamic and electric torques on the fluid. The different ER responses are consistent with the idea that main-chain LCPs are highly prolate, the side-on side chain LCP is moderately prolate, and the end-on side chain LCP is quasi-spherical. A molecular description is obtained by equating eta(on) and eta(off), respectively, to the Miesowicz viscosities eta(c) and eta(b), and using a hydrodynamical model developed by Brochard which de duces that delta eta(c)/delta eta(b) = R-parallel to(4)/R-perpendicular to(4), where R-parallel to and R-perpendicular to are the end-to-end distances of the chain, respectively, parallel and perpendicular to the director.