Projective dimension and regularity of the path ideal of the line graph

TL;DR

This paper investigates algebraic properties of path ideals of line graphs, establishing their sequential Cohen-Macaulayness and deriving explicit formulas for projective dimension, regularity, and depth.

Contribution

It generalizes the notion of path ideals for line graphs and provides exact algebraic invariants, including formulas for projective dimension, regularity, and depth.

Findings

Quotient rings of these path ideals are sequentially Cohen-Macaulay.

Exact formulas for projective dimension and regularity are derived.

The paper also provides formulas for the depth of these ideals.

Abstract

By generalizing the notion of the path ideal of a graph, we study some algebraic properties of some path ideals associated to a line graph. We show that the quotient ring of these ideals are always sequentially Cohen-Macaulay and also provide some exact formulas for the projective dimension and the regularity of these ideals. As some consequences, we give some exact formulas for the depth of these ideals.