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 the experimental one. The results show that there is an obvious linear relationship between $P_{alpha}$ and the product of valance protons (holes) $N_p$ and valance 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 closed shells 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 two different microscopic formulas. We find that the $P_{alpha}$ are also related to $N_pN_n$. 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/ PhysRevC 96.024318} {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 nuclei around those closed shells is model independent. It may be caused by the effect of the valence protons (holes) and valence neutrons (holes) around the shell closures. 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 agree with the experimental data well.
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