  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 \emph{subdivision-vertex join} of $G_1$ and $G_2$,
denoted by $G_1\dot{\vee}G_2$, is the graph obtained from $\mathcal{S}(G_1)$
and $G_2$ by joining every vertex of $V(G_1)$ with every vertex of $V(G_2)$.
The \emph{subdivision-edge join} of $G_1$ and $G_2$, denoted by
$G_1\underline{\vee}G_2$, is the graph obtained from $\mathcal{S}(G_1)$ and
$G_2$ by joining every vertex of $I(G_1)$ with every vertex of $V(G_2)$, 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 $G_1\dot{\vee}G_2$ (respectively,
$G_1\underline{\vee}G_2$) for a regular graph $G_1$ and an arbitrary graph
$G_2$, 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. We also give the number of the spanning trees and the
Kirchhoff index of $G_1\dot{\vee}G_2$ (respectively, $G_1\underline{\vee}G_2$)
for a regular graph $G_1$ and an arbitrary graph $G_2$.
