Path Ideals of Weighted Graphs

TL;DR

This paper introduces weighted path ideals for graphs, generalizing previous unweighted and weighted edge ideals, and characterizes their algebraic properties such as primary decomposition and Cohen-Macaulayness.

Contribution

It defines the weighted r-path ideal for graphs and provides explicit descriptions of primary decompositions and Cohen-Macaulay conditions for specific graph classes.

Findings

Explicit primary decompositions over a field

Characterization of Cohen-Macaulayness for trees and complete graphs

Generalization of existing unweighted and weighted graph ideals

Abstract

We introduce and study the weighted $r$-path ideal of a weighted graph $G_\omega$, which is a common generalization of Conca and De Negri's $r$-path ideal for unweighted graphs and Paulsen and Sather-Wagstaff's edge ideal of the weighted graph. Over a field, we explicitly describe primary decompositions of these ideals, and we characterize Cohen-Macaulayness of these ideals for trees (with arbitrary $r$) and complete graphs (for $r=2$).