  In this paper, we shall give a new upper diameter estimate for complete
Riemannian manifolds in the case that the Bakry-\'Emery Ricci curvature has a
positive lower bound and the norm of the potential function has an upper bound.
Our diameter estimate improves previous ones obtained by Wei and Wylie (J.
Differential Geom. 83, 377--405, 2009) and Limoncu (Math. Z. 271, 715--722,
2012). As an application, we shall give an upper diameter bound for compact
Ricci solitons in terms of the maximum value of the scalar curvature. By using
such a diameter bound, we shall provide some new sufficient conditions for
four-dimensional compact Ricci solitons to satisfy the Hitchin-Thorpe
inequality.
