  Let $G=(V(G),E(G))$ be a graph with vertex set $V(G)$ and edge set $E(G)$.
The subdivision graph $\mathcal{S}(G)$ of a graph $G$ is the graph obtained by
inserting a new vertex into every edge of $G$. Let $G_1$ and $G_2$ be two
vertex disjoint graphs. The subdivision-vertex neighbourhood corona of $G_1$
and $G_2$, denoted by $G_1 \boxdot G_2$, is the graph obtained from
$\mathcal{S}(G_1)$ and $|V(G_1)|$ copies of $G_2$, all vertex disjoint, and
joining the neighbours of the $i$th vertex of $V(G_1)$ to every vertex in the
$i$th copy of $G_2$. The subdivision-edge neighbourhood corona of $G_1$ and
$G_2$, denoted by $G_1 \boxminus G_2$, is the graph obtained from
$\mathcal{S}(G_1)$ and $|I(G_1)|$ copies of $G_2$, all vertex disjoint, and
joining the neighbours of the $i$th vertex of $I(G_1)$ to every vertex in the
$i$th copy of $G_2$, where $I(G_1)$ is the set of inserted vertices of
$\mathcal{S}(G_1)$. In this paper we determine the adjacency spectra, the
Laplacian spectra and the signless Laplacian spectra of $G_1\boxdot G_2$
(respectively, $G_1\boxminus G_2$) in terms of the corresponding spectra of
$G_1$ and $G_2$. As applications, these results enable us to construct
infinitely many pairs of cospectral graphs, and using the results on the
Laplacian spectra of subdivision-vertex neighbourhood coronae, new families of
expander graphs are constructed from known ones.
