  The discovery of pulsations in some low-mass white dwarfs, including the
so-called extremely low-mass white dwarfs, has opened the unprecedented
opportunity of probing the internal structure of these ancient stars. We
present a detailed adiabatic pulsational study of these stars based on a new
set of He-core white-dwarf models with masses ranging from $0.1554$ to $0.4352
M_{\odot}$ derived by computing the non-conservative evolution of a binary
system consisting of an initially $1 M_{\odot}$ ZAMS star and a $1.4 M_{\odot}$
neutron star. We computed adiabatic radial ($\ell= 0$) and non-radial ($\ell=
1, 2$) $p$ and $g$ modes to assess the dependence of the pulsational properties
of these objects on stellar parameters such as the stellar mass and the
effective temperature, as well as the effects of element diffusion. We found
that for white dwarf models with masses below $\sim 0.18 M_{\odot}$, $g$ modes
mainly probe the core regions and $p$ modes the envelope, therefore pulsations
offer the opportunity of constraining both the core and envelope chemical
structure of these stars via asteroseismology. For models with $M_* \gtrsim
0.18 M_{\odot}$, on the other hand, $g$ modes are very sensitive to the He/H
compositional gradient and therefore can be used as a diagnostic tool for
constraining the H envelope thickness. Because both types of objects have not
only very distinct evolutionary histories (according to whether the progenitor
stars have experienced CNO-flashes or not), but also have strongly different
pulsation properties, we propose to define white dwarfs with masses below $\sim
0.18 M_{\odot}$ as ELM (extremely low-mass) white dwarfs, and white dwarfs with
$M_* \gtrsim 0.18 M_{\odot}$ as LM (low-mass) white dwarfs.
