Index of Graded Filiform and Quasi Filiform Lie Algebras

The filiform and the quasi-filiform Lie algebras form a special class of nilpotent Lie algebras. The aim of this paper is to compute the index and provide regular vectors of this two class of nilpotent Lie algebras. we consider the graded filiform Lie algebras $L_{n}$, $Q_{n}$, the $n$-dimensional filiform Lie algebras for $n$ $<8$, also the graded quasi-filiform Lie algebras and finally Lie algebras whose nilradical is $Q_{2n}$.