# Trung's Construction and the Charney-Davis Conjecture

## TL;DR

This paper introduces Trung's construction for graphs, proves its properties related to well-coveredness and Gorenstein conditions, and explores its relation to the Charney-Davis conjecture, especially for planar graphs.

## Contribution

It establishes conditions under which Trung's construction preserves certain graph properties and links these to the Charney-Davis conjecture, providing new insights into Gorenstein planar graphs.

## Key findings

- Trung's construction preserves well-covered, W2, and Gorenstein properties.
- A formula for the independence polynomial of the constructed graph is provided.
- Every Gorenstein planar graph with girth at least four satisfies 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.