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In this work, we systematically study the $\alpha$ decay preformation factors$P_{\alpha}$ and $\alpha$ decay half-lives of 152 nuclei around $Z$ = 82, $N$ =126 closed shells based on a generalized liquid drop model while $P_{\alpha}$is extracted from the ratio of the calculated $\alpha$ decay half-life to theexperimental one. The results show that there is an obvious linear relationshipbetween $P_{\alpha}$ and the product of valance protons (holes) $N_p$ andvalance neutrons (holes) $N_n$. At the same time, we extract the $$P_{\alpha}$values of even-even nuclei around $\emph{Z}$ = 82, $\emph{N}$ = 126 closedshells from the work of Sun \textit{et al.} [\href{https://doi.org/10.1088/1361-6471/aac981} {J. Phys. G: Nucl. Part. Phys.$\bm{45}$, 075106 (2018)}], in which the $P_{\alpha}$ can be calculated by twodifferent microscopic formulas. We find that the $P_{\alpha}$ are also relatedto $N_pN_n$. Combining with our previous works [Sun \textit{et al.}, \href{https://doi.org/10.1103/PhysRevC.94.024338} {Phys. Rev. C $\bm{94}$, 024338(2016)}; Deng \textit{et al.}, \href {https://doi.org/10.1103/ PhysRevC96.024318} {ibid. $\bm{96}$, 024318 (2017)}; Deng \textit{et al.}, \href{https://doi.org/10.1103/PhysRevC.97.044322} {ibid. $\bm{97}$, 044322 (2018)}]and the work of Seif \textit{et al.} [\href{http://dx.doi.org/10.1103/PhysRevC.84.064608}{Phys. Rev. C $\bm{84}$, 064608(2011)}], we suspect that this phenomenon of linear relationship for the nucleiaround those closed shells is model independent. It may be caused by the effectof the valence protons (holes) and valence neutrons (holes) around the shellclosures. Finally, using the formula obtained by fitting the $P_{\alpha}$calculated by the generalized liquid drop model (GLDM), we calculate the$\alpha$ decay half-lives of these nuclei. The calculated results are agreewith the experimental data well.

Systematic study of the $\alpha$ decay preformation factor of nuclei  around the $\boldsymbol{Z=82}$, $\boldsymbol{N=126}$ shell closures within a  generalized liquid drop model