\begin{document}$ P_c $\end{document} is discussed in terms of the \begin{document}$ N_p N_n $\end{document} scheme to study the influence of the valence proton-neutron interaction and shell effect on cluster radioactivity. It is found that \begin{document}$ \log_{10}{P_c} $\end{document} is linearly related to \begin{document}$ N_p N_n $\end{document}. This is in agreement with a recent study [L. Qi et al., Phys. Rev. C 108, 014325 (2023)], in which \begin{document}$ \log_{10}{P_c} $\end{document}, obtained using different theoretical models and treatment methods than those used in this study, also had a linear relationship with \begin{document}$ N_p N_n $\end{document}. Combined with the work by Qi et al., this study suggests that the linear relationship between \begin{document}$ \log_{10}{P_c} $\end{document} and \begin{document}$ N_p N_n $\end{document} is model-independent and both the shell effect and valence proton-neutron interaction play essential roles in cluster radioactivity. An analytical formula is proposed to calculate the cluster preformation factor based on the \begin{document}$ N_p N_n $\end{document} scheme. In addition, the cluster preformation factors and the cluster radioactivity half-lives of some heavy nuclei are predicted, which can provide a reference for future experiments."> Systematic study of cluster radioactivity within the generalized liquid drop model -
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