  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.
