TESS giants transiting giants V - two hot Jupiters orbiting red giant hosts

In this work, we present the discovery and confirmation of two hot Jupiters orbiting red giant stars, TOI-4377 b and TOI-4551 b, observed by Transiting Exoplanet Survey Satellite in the Southern ecliptic hemisphere and later followed-up with radial-velocity (RV) observations. For TOI-4377 b, we report a mass of $0.957^{+0.089}_{-0.087} \ M_\mathrm{J}$ and a inflated radius of 1.348 ± 0.081 R_J orbiting an evolved intermediate-mass star (1.36 M_⊙ and 3.52 R_⊙; TIC 394918211) on a period of of 4.378 d. For TOI-4551 b, we report a mass of 1.49 ± 0.13 M_J and a radius that is not obviously inflated of $1.058^{+0.110}_{-0.062} \ R_\mathrm{J}$, also orbiting an evolved intermediate-mass star (1.31 M_⊙ and 3.55 R_⊙; TIC 204650483) on a period of 9.956 d. We place both planets in context of known systems with hot Jupiters orbiting evolved hosts, and note that both planets follow the observed trend of the known stellar incident flux-planetary radius relation observed for these short-period giants. Additionally, we produce planetary interior models to estimate the heating efficiency with which stellar incident flux is deposited in the planet's interior, estimating values of $1.91 \pm 0.48~{{\ \rm per\ cent}}$ and $2.19 \pm 0.45~{{\ \rm per\ cent}}$ for TOI-4377 b and TOI-4551 b, respectively. These values are in line with the known population of hot Jupiters, including hot Jupiters orbiting main-sequence hosts, which suggests that the radii of our planets have re-inflated in step with their parent star's brightening as they evolved into the post-main sequence. Finally, we evaluate the potential to observe orbital decay in both systems.