Gradings and graded identities of null-filiform Leibniz algebras

We classify gradings on null-filiform Leibniz algebras up to equivalence over arbitrary fields. Furthermore, we provide a basis for the graded identities and determine a basis of the relatively free algebra. As a consequence, we establish that the ideal of all graded identities of null-filiform Leibniz algebras satisfy the Specht property. Finally, we extend these results to infinite-dimensional analogs of null-filiform Leibniz algebras.