  Our aim in this article is to give a lower bound of the diameter of a compact
gradient $\rho$-Einstein soliton satisfying some given conditions. We have also
deduced some conditions of the gradient $\rho$-Einstein soliton with bounded
Ricci curvature to become non-shrinking and non-expanding. Further, we have
proved that a complete non-compact gradient shrinking or expanding Schouten
soliton with non-constant potential and a boundedness condition on scalar
curvature must be non-parabolic.
