# Algebraic properties of toric rings of graphs

## TL;DR

This paper studies algebraic properties of toric rings derived from graphs, focusing on Cohen-Macaulayness and invariants like regularity and projective dimension, by analyzing induced subgraphs.

## Contribution

It introduces methods to relate algebraic invariants of the toric ring of a graph to those of its induced subgraphs, advancing understanding of their algebraic structure.

## Key findings

- Established connections between invariants of $k[G]$ and induced subgraphs.
- Provided criteria for Cohen-Macaulayness of toric rings of graphs.
- Analyzed bounds for Castelnuovo-Mumford regularity and projective dimension.

## Abstract

Let $G = (V,E)$ be a simple graph. We investigate the Cohen-Macaulayness and algebraic invariants, such as the Castelnuovo-Mumford regularity and the projective dimension, of the toric ring $k[G]$ via those of toric rings associated to induced subgraphs of $G$.