  We study some spectral properties of Volterra-type integral operators $V_g$
and $I_g$ with holomorphic symbol $g$ on the Fock--Sobolev spaces
$\mathcal{F}_{\psi_m}^p$. We showed that $V_g$ is bounded on
$\mathcal{F}_{\psi_m}^p$ if and only if $g$ is a complex polynomial of degree
not exceeding two, while compactness of $V_g$ is described by degree of $g$
being not bigger than one. We also identified all those positive numbers $p$
for which the operator $V_g$ belongs to the Schatten $\mathcal{S}_p$ classes.
Finally, we characterize the spectrum of $V_g$ in terms of a closed disk of
radius twice the coefficient of the highest degree term in a polynomial
expansion of $g$.
