Values and bounds for depth and Stanley depth of some classes of edge ideals

Naeem Ud Din; Muhammad Ishaq; Zunaira Sajid; Naeem Ud Din; Muhammad Ishaq; Zunaira Sajid

doi:10.3934/math.2021496

AIMS Mathematics

2021, Volume 6, Issue 8: 8544-8566. doi: 10.3934/math.2021496

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Research article Special Issues

Values and bounds for depth and Stanley depth of some classes of edge ideals

Department of Mathematics, School of Natural Sciences, National University of Sciences and Technology, H-12, Islamabad, Pakistan

Received: 18 March 2021 Accepted: 23 May 2021 Published: 07 June 2021
MSC : Primary 13C15; Secondary 13P10, 13F20

In this paper we study depth and Stanley depth of the quotient rings of the edge ideals associated with the corona product of some classes of graphs with arbitrary non-trivial connected graph $ G $. These classes include caterpillar, firecracker and some newly defined unicyclic graphs. We compute formulae for the values of depth that depend on the depth of the quotient ring of the edge ideal $ I(G) $. We also compute values of depth and Stanley depth of the quotient rings associated with some classes of edge ideals of caterpillar graphs and prove that both of these invariants are equal for these classes of graphs.
- depth,
- Stanley depth,
- Stanley decomposition,
- monomial ideal,
- edge ideal
Citation: Naeem Ud Din, Muhammad Ishaq, Zunaira Sajid. Values and bounds for depth and Stanley depth of some classes of edge ideals[J]. AIMS Mathematics, 2021, 6(8): 8544-8566. doi: 10.3934/math.2021496

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Abstract

In this paper we study depth and Stanley depth of the quotient rings of the edge ideals associated with the corona product of some classes of graphs with arbitrary non-trivial connected graph $ G $. These classes include caterpillar, firecracker and some newly defined unicyclic graphs. We compute formulae for the values of depth that depend on the depth of the quotient ring of the edge ideal $ I(G) $. We also compute values of depth and Stanley depth of the quotient rings associated with some classes of edge ideals of caterpillar graphs and prove that both of these invariants are equal for these classes of graphs.

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