The DM_IGM − z relation for fast radio bursts (FRBs). The scatter star points are 23 localized FRBs. DM_IGM values are derived by deducing the DM_MW, DM_halo, and DM_host from DM_obs. DM_MW values are deduced from the NE2001 model, and DM_halo values are estimated to be 65 pc cm⁻³. We use the median value of DM_host at different redshifts according to Equation (5). The blue line is the result derived from IllustrisTNG 300 cosmological simulation and the green shaded region is the 95% confidence level. The red line represents the theoretical model DM_IGM (Equation (2)) assuming f _IGM = 0.82, Ω _b = 0.0488 and H ₀ = 70 km s⁻¹ Mpc⁻¹. Due to FRB 210117 (red point) deviating from the 95% confidence region of DM_IGM, we exclude it in the following analysis.