Trung's Construction and the Charney–Davis Conjecture.

We consider a construction by which we obtain a simple graph Tr (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 Tr (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 Tr (H , v) and investigate when the independence complex of Tr (H , v) satisfies the Charney–Davis conjecture. As a consequence of our results, we show that the independence complex of every Gorenstein planar graph with girth at least four satisfies the Charney–Davis conjecture. [ABSTRACT FROM AUTHOR]