Algebraic invariants of edge ideals from strong products of paths and cycles

Ahtsham Ul Haq; Muhammad Usman Rashid; Muhammad Ishaq; Ahtsham Ul Haq; Muhammad Usman Rashid; Muhammad Ishaq

doi:10.3934/math.20251303

AIMS Mathematics

2025, Volume 10, Issue 12: 29650-29663. doi: 10.3934/math.20251303

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Algebraic invariants of edge ideals from strong products of paths and cycles

1.
Department of Mathematics, School of Natural Sciences, National University of Sciences and Technology, Sector H-12, Islamabad 44000, Pakistan
2.
Department of Sciences and Humanities, National University of Computer and Emerging Sciences, Islamabad 44000, Pakistan

Received: 29 September 2025 Revised: 27 November 2025 Accepted: 04 December 2025 Published: 16 December 2025
MSC : 05C69, 05C70, 13D02, 13E40

This paper investigates algebraic invariants of edge ideals associated with families of graphs constructed as the strong product of a path or cycle with the complete graph $ K_m $, namely $ \mathscr{P}_\gamma = P_\gamma \boxtimes K_m $ and $ \mathscr{C}_\gamma = C_\gamma \boxtimes K_m $. For an edge ideal $ I(\mathbb{G}) \subset \Re $ in a polynomial ring over a field $ \mathbb{K} $, we derive explicit combinatorial formulas for key homological and ring-theoretic invariants of the quotient ring $ \Re/I(\mathbb{G}) $. These include Castelnuovo-Mumford regularity, depth, Stanley depth, projective dimension, and Krull dimension. Furthermore, we characterize all Cohen–Macaulay graphs in defined families, providing a complete classification where $ \Re/I(\mathbb{G}) $ is Cohen–Macaulay.
- edge ideal,
- depth,
- projective dimension,
- stanley depth,
- Castelnuovo-Mumford regularity,
- Krull dimension,
- Cohen–Macaulay rings
Citation: Ahtsham Ul Haq, Muhammad Usman Rashid, Muhammad Ishaq. Algebraic invariants of edge ideals from strong products of paths and cycles[J]. AIMS Mathematics, 2025, 10(12): 29650-29663. doi: 10.3934/math.20251303

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Abstract

This paper investigates algebraic invariants of edge ideals associated with families of graphs constructed as the strong product of a path or cycle with the complete graph $ K_m $, namely $ \mathscr{P}_\gamma = P_\gamma \boxtimes K_m $ and $ \mathscr{C}_\gamma = C_\gamma \boxtimes K_m $. For an edge ideal $ I(\mathbb{G}) \subset \Re $ in a polynomial ring over a field $ \mathbb{K} $, we derive explicit combinatorial formulas for key homological and ring-theoretic invariants of the quotient ring $ \Re/I(\mathbb{G}) $. These include Castelnuovo-Mumford regularity, depth, Stanley depth, projective dimension, and Krull dimension. Furthermore, we characterize all Cohen–Macaulay graphs in defined families, providing a complete classification where $ \Re/I(\mathbb{G}) $ is Cohen–Macaulay.

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