Depth and Stanley depth of the edge ideals of the strong product of some graphs

TL;DR

This paper investigates the depth and Stanley depth of edge ideals from the strong product of graphs, providing exact formulas for specific cases and bounds for general cases.

Contribution

It introduces formulas and bounds for the depth and Stanley depth of edge ideals of strong product graphs, focusing on paths and cycles.

Findings

Exact formulas for depth and Stanley depth when both graphs are paths.

Exact formulas for depth and Stanley depth when one graph is a path and the other a cycle.

Sharp upper bounds for depth and Stanley depth in general cases.

Abstract

In this paper we study depth and Stanley depth of the edge ideals and quotient rings of the edge ideals, associated to classes of graphs obtained by taking the strong product of two graphs. We consider the cases when either both graphs are arbitrary paths or one is an arbitrary path and the other is an arbitrary cycle. We give exact formulae for values of depth and Stanley depth for some sub classes. We also give some sharp upper bounds for depth and Stanley depth in the general cases.