  We consider a construction by which we obtain a simple graph
$\mathrm{T}(H,v)$ from a simple graph $H$ and a non-isolated vertex $v$ of $H$.
We call this construction "Trung's construction". We prove that
$\mathrm{T}(H,v)$ is well-covered, W$_2$ or Gorenstein if and only if $H$ is
so. Also we present a formula for computing the independence polynomial of
$\mathrm{T}(H,v)$ and investigate when $\mathrm{T}(H,v)$ satisfies the
Charney-Davis conjecture. As a consequence of our results, we show that every
Gorenstein planar graph with girth at least four, satisfies the Charney-Davis
conjecture.
