Context : The number of detected extremely low mass ( ELM ) white dwarf stars has increased drastically in recent year thanks to the results of many surveys .
In addition , some of these stars have been found to exhibit pulsations , making them potential targets for asteroseismology .

Aims : We provide a fine and homogeneous grid of evolutionary sequences for helium ( He ) core white dwarfs for the whole range of their expected masses ( 0.15 \lesssim M _ { * } / M _ { \sun } \lesssim 0.45 ) , including the mass range for ELM white dwarfs ( M _ { * } / M _ { \sun } \lesssim 0.20 ) .
The grid is appropriate for mass and age determination of these stars , as well as to study their adiatabic pulsational properties .

Methods : White dwarf sequences have been computed by performing full evolutionary calculations that consider the main energy sources and processes of chemical abundance changes during white dwarf evolution .
Realistic initial models for the evolving white dwarfs have been obtained 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 for various initial orbital periods .
To derive cooling ages and masses for He-core white dwarf we perform a least square fitting of the M ( T _ { eff } ,g ) and { Age } ( T _ { eff } ,g ) relations provided by our sequences by using a scheme that takes into account the time spent by models in different regions of the T _ { eff } - g plane .
This is particularly useful when multiple solutions for cooling age and mass determinations are possible in the case of CNO-flashing sequences .
We also explore in a preliminary way the adiabatic pulsational properties of models near the critical mass for the development of CNO flashes ( \sim 0.2 M _ { \odot } ) .
This is motivated by the discovery of pulsating white dwarfs with stellar masses near this threshold value .

Results : We obtain reliable and homogeneous mass and cooling age determinations for 58 very low-mass white dwarfs , including 3 pulsating stars .
Also , we find substantial differences in the period spacing distributions of g -modes for models with stellar masses near \sim 0.2 M _ { \odot } , which could be used as a seismic tool to distinguish stars that have undergone CNO flashes in their early cooling phase from those that have not .
Finally , for an easy application of our results , we provide a reduced grid of values useful to obtain masses and ages of He-core white dwarf .

Conclusions :