Multipliers of nilpotent Lie superalgebras

In this paper, first we prove that all finite dimensional special Heisenberg Lie superalgebras with even center have same dimension, say $(2m+1\mid n)$ for some non-negative integers $m,n$ and are isomorphism with them. Further, for a nilpotent Lie superalgebra $L$ of dimension $(m\mid n)$ and $\dim (L') = (r\mid s)$ with $r+s \geq 1$, we find the upper bound $\dim \mathcal{M}(L)\leq \frac{1}{2}\left[(m + n + r + s - 2)(m + n - r -s -1) \right] + n + 1$, where $\mathcal{M}(L)$ denotes the Schur multiplier of $L$. Moreover, if $(r, s) =(1, 0)\; (\mathrm{respectively}\; (r,s) = (0,1))$, then the equality holds if and only if $L \cong H(1,0) \oplus A_{1}\; (\mathrm{respectively}\; H(0,1) \oplus A_{2})$, where $A_{1}$ and $A_{2}$ are abelian Lie superalgebras with $\dim A_{1}=(m-3 \mid n), \dim A_{2}=(m-1 \mid n-1)$ and $H(1,0), H(0,1)$ are special Heisenberg Lie superalgebras of dimension $3$ and $2$ respectively.