Let \(\lambda\) be a radical property in the category of associative rings and \(G\) a group. By means of the smash product a corresponding radical property \(\lambda_{\text{ref}}\) is defined in the category of associative \(G\)-graded rings. The authors describe these radicals and the relationship with the corresponding classical graded radicals for the Jacobson, prime, strongly prime, Levitzki, Brown-McCoy and von Neuman regular radicals. Some examples are given for the cases when \(\lambda_{\text{ref}}\) and the corresponding graded radical are not the same.