Context : Many low-mass white dwarfs with masses M _ { * } / M _ { \odot } \lesssim 0.45 , including the so-called extremely low-mass white dwarfs ( M _ { * } / M _ { \odot } \lesssim 0.20 - 0.25 ) , have recently been discovered in the field of our Galaxy through dedicated photometric surveys .
The subsequent discovery of pulsations in some of them has opened the unprecedented opportunity of probing the internal structure of these ancient stars .

Aims : We present a detailed adiabatic pulsational study of these stars based on full evolutionary sequences derived from binary star evolution computations .
The main aim of this study is to provide a detailed theoretical basis of reference for interpreting present and future observations of variable low-mass white dwarfs .

Methods : Our pulsational analysis is 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 .

Results : 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 .

Conclusions :