  We prove first existence of a classical solution to a class of parabolic
problems with unbounded coefficients on metric star graphs subject to
Kirchhoff-type conditions. The result is applied to the Ornstein--Uhlenbeck and
the harmonic oscillator operators on metric star graphs. We give an explicit
formula for the associated Ornstein--Uhlenbeck semigroup and give the unique
associated invariant measure. We show that this semigroup inherits the
regularity properties of the classical Ornstein--Uhlenbeck semigroup on
$\mathbb R$.
