Structural properties of hyperideal-based zero-divisor graphs over noncommutative hyperrings

Amal S. Alali; Kajal Rani; Junaid Nisar; Shabir Ahmad Mir; Shalu Rani; Amal S. Alali; Kajal Rani; Junaid Nisar; Shabir Ahmad Mir; Shalu Rani

doi:10.3934/math.2026631

AIMS Mathematics

2026, Volume 11, Issue 6: 15361-15375. doi: 10.3934/math.2026631

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Structural properties of hyperideal-based zero-divisor graphs over noncommutative hyperrings

1.
Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P. O. Box 84428, Riyadh 11671, Saudi Arabia
2.
Department of Mathematics, Lovely Professional University, Punjab 144411, India
3.
Symbiosis Institute of Technology, Pune Campus, Symbiosis International (Deemed University), Pune, India
4.
Department of Mathematics, Aligarh Muslim University, Aligarh-202002, India
5.
Department of Mathematics, Panjab University, Chandigarh, India

Received: 13 March 2026 Revised: 11 May 2026 Accepted: 18 May 2026 Published: 01 June 2026
MSC : 05C25, 14A22, 15A18;, 16Y99

In this paper, we study hyperideal-based zero-divisor graphs $ \Gamma_I(R) $ associated with noncommutative hyperrings. We investigate fundamental structural properties of these graphs under the given adjacency condition, including connectivity, diameter bounds, and coloring parameters. For finite hyperrings, we characterize connectedness of $ \Gamma_I(R) $ using the rank of the Laplacian matrix. We also establish general bounds on the diameter and identify conditions under which the diameter is at most two. Furthermore, we introduce algebraic measures such as the extended chromatic number and an asymmetry index to capture effects arising from noncommutativity. Several results illustrate how asymmetry in hypermultiplication influences graph structure, including possible differences between classical and extended coloring.
- zero-divisor graphs,
- hyperideal,
- non-commutative hyperring,
- nonassociative structure
Citation: Amal S. Alali, Kajal Rani, Junaid Nisar, Shabir Ahmad Mir, Shalu Rani. Structural properties of hyperideal-based zero-divisor graphs over noncommutative hyperrings[J]. AIMS Mathematics, 2026, 11(6): 15361-15375. doi: 10.3934/math.2026631

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Abstract

In this paper, we study hyperideal-based zero-divisor graphs $ \Gamma_I(R) $ associated with noncommutative hyperrings. We investigate fundamental structural properties of these graphs under the given adjacency condition, including connectivity, diameter bounds, and coloring parameters. For finite hyperrings, we characterize connectedness of $ \Gamma_I(R) $ using the rank of the Laplacian matrix. We also establish general bounds on the diameter and identify conditions under which the diameter is at most two. Furthermore, we introduce algebraic measures such as the extended chromatic number and an asymmetry index to capture effects arising from noncommutativity. Several results illustrate how asymmetry in hypermultiplication influences graph structure, including possible differences between classical and extended coloring.

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