Depth and Stanley depth of the path ideal associated to an $n$-cyclic graph

TL;DR

This paper calculates the depth and Stanley depth of quotient rings from path ideals of length 3 in n-cyclic graphs, providing formulas and bounds based on modular arithmetic conditions.

Contribution

It offers new formulas and bounds for depth and Stanley depth of path ideals in cyclic graphs, expanding understanding of their algebraic properties.

Findings

Exact formulas for depth when n ≠ 1 mod 4

Tight bounds for Stanley depth when n ≡ 0,3 mod 4

Formulas for depth and Stanley depth of path ideals of lengths n-1 and n

Abstract

We compute the depth and Stanley depth for the quotient ring of the path ideal of length $3$ associated to a $n$-cyclic graph, given some precise formulas for depth when $n\not\equiv 1\,(\mbox{mod}\ 4)$, tight bounds when $n\equiv 1\,(\mbox{mod}\ 4)$ and for Stanley depth when $n\equiv 0,3\,(\mbox{mod}\ 4)$, tight bounds when $n\equiv 1,2\,(\mbox{mod}\ 4)$. Also, we give some formulas for depth and Stanley depth of a quotient of the path ideals of length $n-1$ and $n$.