  Given two graphs $G$ and $H$, assume that $\mathscr{C}=\{C_1,C_2,\ldots,
C_q\}$ is a clique cover of $G$ and $U$ is a subset of $V(H)$. We introduce a
new graph operation called the clique cover product, denoted by
$G^{\mathscr{C}}\star H^U$, as follows: for each clique $C_i\in \mathscr{C}$,
add a copy of the graph $H$ and join every vertex of $C_i$ to every vertex of
$U$. We prove that the independence polynomial of $G^{\mathscr{C}}\star H^U$
$$I(G^{\mathscr{C}}\star H^U;x)=I^q(H;x)I(G;\frac{xI(H-U;x)}{I(H;x)}),$$ which
generalizes some known results on independence polynomials of corona and rooted
products of graphs obtained by Gutman and Rosenfeld, respectively. Based on
this formula, we show that the clique cover product of some special graphs
preserves symmetry, unimodality, log-concavity or reality of zeros of
independence polynomials. As applications we derive several known facts in a
unified manner and solve some unimodality conjectures and problems.
