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Let $\bar{G}$ be the simple algebraic supergroup $\mathrm{SL}(m|n)$ or$\mathrm{OSp}(m|2n)$ over $\mathbb{C}$. Let$\mathfrak{g}=\mathrm{Lie}(\bar{G})=\mathfrak{g}_{\bar{0}}\oplus\mathfrak{g}_{\bar{1}}$and let $G=\bar{G}(\mathbb{C})$ where $\mathbb{C}$ is considered as asuperalgebra concentrated in even degree. Suppose $e\in\mathfrak{g}_{\bar{0}}$is nilpotent. We describe the centralizer $\mathfrak{g}^{e}$ of $e$ in$\mathfrak{g}$ and its centre $\mathfrak{z}(\mathfrak{g}^{e})$. In particular,we give bases for $\mathfrak{g}^{e}$, $\mathfrak{z}(\mathfrak{g}^{e})$ and$\left(\mathfrak{z}(\mathfrak{g}^{e})\right)^{G^{e}}$. We also determine thelabelled Dynkin diagram $\varDelta$ with respect to $e$ and subsequentlydescribe the relation between$\left(\mathfrak{z}(\mathfrak{g}^{e})\right)^{G^{e}}$ and $\varDelta$.

Centres of centralizers of nilpotent elements in Lie superalgebras $\mathfrak{sl}(m|n)$ or $\mathfrak{osp}(m|2n)$