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.
