A sample study on the Fermi-LAT spectral variation behavior of blazars

In this study, we analyze the $\sim 16$ yr Fermi-LAT data of 160 blazars consisting of 92 flat-spectrum radio quasars (FSRQs), 64 BL Lacertae type objects (BL Lacs), and 4 blazar candidates of unknown type objects (BCU), and exhibit their spectral variation. By fitting a linear model ($y=kx+b$) to the monthly binned flux-index plots, we find that most of the FSRQs ($\sim 89\mathrm{\%}$) display a harder-when-brighter (HWB) trend ($k>0$), while the majority of BL Lacs ($\sim 64\mathrm{\%}$) exhibit a softer-when-brighter (SWB) trend ($k<0$). By plotting their slope ($k$) versus their variability index and inverse Compton (IC) peak frequency, we find a moderate positive correlation between $k$ and the variability index, and we find a moderate negative correlation between $k$ and the IC peak frequency, which we name as the gamma-ray variability sequence of blazars. Additionally, 7 blazars may not follow the linear variation trend, particularly in PKS 0537-441, AP Librae, and PKS 1510-089. Additional very-high-energy gamma-ray emission components may be the cause of the break in the variation trend.