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The cover product of disjoint graphs 
$G$ and 
$H$ with fixed vertex covers 
$C\left( G \right)$ and 
$C\left( H \right)$, is the graph 
$G\circledast H$ with vertex set 
$V\left( G \right)\cup V\left( H \right)$ and edge set
$$E\left( G \right)\,\cup \,E\left( H \right)\,\cup \,\left\{ \left\{ i,\,j \right\}\,:\,i\,\in \,C\left( G \right),\,j\,\in \,C\left( H \right) \right\}.$$
We describe the graded Betti numbers of $G\circledast H$ in terms of those of $G$ and $H$. As applications we obtain: (i) For any positive integer k there exists a connected bipartite graph $G$ such that 
$\text{reg}\,R/I\left( G \right)\,=\,{{\mu }_{s}}\left( G \right)\,+\,k$, where, 
$I\left( G \right)$ denotes the edge ideal of $G$, 
$\text{reg}\,\text{R/I}\left( G \right)$ is the Castelnuovo–Mumford regularity of 
$\text{R/I}\left( G \right)$ and 
${{\mu }_{s}}\left( G \right)$ is the induced or strong matching number of $G$; (ii)The graded Betti numbers of the complement of a tree depends only upon its number of vertices; (iii)The 
$h$-vector of 
$R/I\left( G\circledast H \right)$ is described in terms of the $h$-vectors of $\text{R/I}\left( G \right)$ and 
$R/I\left( H \right)$. Furthermore, in a different direction, we give a recursive formula for the graded Betti numbers of chordal bipartite graphs.
