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If $\mathfrak{n}$ is a $\mathbb{Z}^d_+$-graded nilpotent finite dimensionalLie algebra over a field of characteristic zero, it is well known that $\dimH^{\ast }(\mathfrak{n})\geq L(p) $ where $p$ is the polynomial associated tothe grading and $L(p)$ is the sum of the absolute values of the coefficients of$p$. From this result Deninger and Singhof derived the Toral Rank Conjecture(TRC) for 2-step nilpotent Lie algebras. An algebraic version of the TRC statesthat $\dim H^{\ast }(\mathfrak{n})\geq 2^{\dim (\mathfrak{z)}} $ for any finitedimensional Lie algebra $\mathfrak{n}$ with center $\mathfrak{z}$.  The TRC is more that 25 years old and remains open even for$\mathbb{Z}^d_+$-graded 3-step nilpotent Lie algebras. Investigating to whatextent the above bound for $\dim H^{\ast }(\mathfrak{n})$ could help to provethe TRC in this case, we considered the following two questions regarding anilpotent Lie algebra $\mathfrak{n}$ with center $\mathfrak{z}$: (A) If$\mathfrak{n}$ admits a $\mathbb{Z}_+^d$-grading  $\mathfrak{n}=\bigoplus_{\alpha\in\mathbb{Z}_+^d} \mathfrak{n}_\alpha$, suchthat its associated polynomial $p$ satisfies $L(p)>2^{\dim\mathfrak{z}}$, doesit admit a grading $\mathfrak{n}=\mathfrak{n}'_{1}\oplus\mathfrak{n}'_{2}\oplus \dots\oplus \mathfrak{n}'_{k}$ such that its associatedpolynomial $p'$ satisfies $L(p')>2^{\dim\mathfrak{z}}$? (B) If $\mathfrak{n}$is $r$-step nilpotent admitting a grading $\mathfrak{n}=\mathfrak{n}_{1}\oplus\mathfrak{n}_{2}\oplus \dots\oplus \mathfrak{n}_{k}$ such that its associatedpolynomial $p$ satisfies $L(p)>2^{\dim\mathfrak{z}}$, does it admit a grading$\mathfrak{n}=\mathfrak{n}'_{1}\oplus \mathfrak{n}'_{2}\oplus \dots\oplus\mathfrak{n}'_{r}$ such that its associated polynomial $p'$ satisfies$L(p')>2^{\dim\mathfrak{z}}$? In this paper we show that the answer to (A) isyes, but the answer to (B) is no.

Some remarks on graded nilpotent Lie algebras and the Toral Rank Conjecture