Integral closure and normality of edge ideals of some edge-weighted graphs

TL;DR

This paper characterizes when the edge ideal of an edge-weighted graph is integrally closed and shows that for certain graph types, integrally closed ideals are also normal.

Contribution

It provides a complete characterization of integrally closed edge ideals for edge-weighted graphs and establishes normality for specific graph classes.

Findings

Characterization of integrally closed edge ideals for edge-weighted graphs

Normality of integrally closed ideals in star, path, and cycle graphs

Conditions under which integrally closed ideals are normal

Abstract

Let $G_\omega$ be an edge-weighted simple graph. In this paper, we give a complete characterization of the graph $G_\omega$ whose edge ideal $I(G_\omega)$ is integrally closed. We also show that if $G_\omega$ is an edge-weighted star graph, a path or a cycle, and $I(G_\omega)$ is integrally closed, then $I(G_\omega)$ is normal.