The weak Lefschetz properties of artinian monomial algebras associated to certain tadpole graphs

TL;DR

This paper classifies certain tadpole graphs based on whether their associated artinian monomial algebras possess the weak Lefschetz property, linking graph structure to algebraic properties.

Contribution

It provides a classification of tadpole graphs for which the associated algebra has or lacks the weak Lefschetz property, a novel connection between graph theory and algebra.

Findings

Identifies specific tadpole graphs with the weak Lefschetz property

Determines conditions under which the property fails

Establishes a link between graph structure and algebraic behavior

Abstract

Given a simple graph $G$, the artinian monomial algebra associated to $G$, denoted by $A(G)$, is defined by the edge ideal of $G$ and the squares of the variables. In this article, we classify some tadpole graphs $G$ for which $A(G)$ has or fails the weak Lefschetz property.