  In recent years there has been a surge of interest in what has come to be
known as the `universal relations' between various global properties of neutron
stars. These universal relations are equation of state independent relations
between quantities such as the moment of inertia $I$, the tidal deformability
or Love number $\lambda$, and the quadrupole $Q$ (I-Love-Q relations), or the
relativistic multipole moments (3-hair relations). While I-Love-Q relations
have been studied extensively in both general relativity and various
alternatives, 3-hair relations have been studied only in general relativity.
Recent progress on the definition of the multipole moments of a compact object
in the case of scalar-tensor theories allows for the study of 3-hair relations
in modified theories of gravity. Specifically, the aim of this work is to study
them for scalarized stars in scalar-tensor theories with a massless scalar
field that admit spontaneous scalarization. We find that the 3-hair relations
between the mass and angular momentum moments that hold in general relativity
hold for scalarized stars as well. The scalar moments also exhibit a universal
behaviour, which is equation of state independent within one specific theory,
but differs between different theories. Combining astrophysical observations
one can in principle measure the different properties of scalarized neutron
star and tell different theories apart.
