Trung's Construction and the Charney-Davis Conjecture

Abstract

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.

References in corpus (1)

• Chordality of Clutters with Vertex Decomposable Dual and Ascent of Clutters

Cited by in corpus (4)

• Gorenstein and Cohen-Macaulay Matching Complexes
• Some Combinatorial Characterizations of Gorenstein Graphs with Independence Number Less than Four
• Well-covered Unitary Cayley Graphs of Matrix Rings over Finite Fields and Applications
• On Gorenstein Circulant Graphs and Gorenstein SQC Graphs