  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 TESS 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 $1.348 \pm 0.081 \ R_\mathrm{J}$
orbiting an evolved intermediate-mass star ($1.36 \ \mathrm{M}_\odot$, $3.52 \
\mathrm{R}_\odot$; TIC 394918211) on a period of of $4.378$ days. For TOI-4551
b we report a mass of $1.49 \pm 0.13 \ M_\mathrm{J}$ 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 ($1.31 \ \mathrm{M}_\odot$, $3.55 \
\mathrm{R}_\odot$; TIC 204650483) on a period of $9.956$ days. 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 $1.91 \pm 0.48\%$ and $2.19 \pm 0.45\%$ 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 reinflated 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.
