Cohen-Macaulay edge-weighted graphs of girth $5$ or greater

TL;DR

This paper classifies Cohen-Macaulay edge-weighted graphs with girth at least 5, expanding the understanding of their structure in algebraic graph theory.

Contribution

It provides a complete classification of Cohen-Macaulay edge-weighted graphs for graphs with girth 5 or more, a significant extension in the field.

Findings

Complete classification of Cohen-Macaulay edge-weighted graphs with girth ≥ 5

Identification of structural properties necessary for Cohen-Macaulayness

Extension of existing classifications to graphs with larger girth

Abstract

Let $G_\omega$ be an edge-weighted graph whose underlying graph is $G$. In this paper, we enlarge the class of Cohen-Macaulay edge-weighted graphs $G_\omega$ by classifying completely them when the graph $G$ has girth $5$ or greater.