On the multiplicative Zagreb indices of zero-divisor graphs of finite commutative reduced rings

R. Sankari Alias Deepa; R. Gurusamy; S. Arockiaraj; Yilun Shang; R. Sankari Alias Deepa; R. Gurusamy; S. Arockiaraj; Yilun Shang

doi:10.3934/math.20261034

AIMS Mathematics

2026, Volume 11, Issue 8: 25814-25838. doi: 10.3934/math.20261034

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Research article

On the multiplicative Zagreb indices of zero-divisor graphs of finite commutative reduced rings

1.
Madurai Kamaraj University, Madurai 625021, Tamil Nadu, India
2.
Department of Mathematics, Mepco Schlenk Engineering College, Sivakasi 626005, Tamil Nadu, India
3.
Department of Mathematics, Government Arts & Science College, Sivakasi 626124, Tamil Nadu, India
4.
School of Computer Science, Northumbria University, Newcastle NE1 8ST, UK

Received: 18 June 2026 Revised: 04 August 2026 Accepted: 13 August 2026 Published: 20 August 2026
MSC : 05C07, 05C25, 13A99, 13M99

Topological indices serve as key numerical invariants for characterizing graph structures and play an important role in chemical graph theory. In this work, we investigated degree-based topological indices of zero-divisor graphs arising from finite commutative reduced rings. Closed-form expressions were established for the Narumi–Katayama index and the first and second multiplicative Zagreb indices for this class of rings. The results obtained here extend and refine earlier studies on degree-based indices of zero-divisor graphs, and we additionally identify and correct certain inaccuracies present in previous analyses by other researchers. Furthermore, we provide Python-based computations to validate the theoretical findings, accompanied by comparative numerical and graphical illustrations.
Citation: R. Sankari Alias Deepa, R. Gurusamy, S. Arockiaraj, Yilun Shang. On the multiplicative Zagreb indices of zero-divisor graphs of finite commutative reduced rings[J]. AIMS Mathematics, 2026, 11(8): 25814-25838. doi: 10.3934/math.20261034

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Abstract

Topological indices serve as key numerical invariants for characterizing graph structures and play an important role in chemical graph theory. In this work, we investigated degree-based topological indices of zero-divisor graphs arising from finite commutative reduced rings. Closed-form expressions were established for the Narumi–Katayama index and the first and second multiplicative Zagreb indices for this class of rings. The results obtained here extend and refine earlier studies on degree-based indices of zero-divisor graphs, and we additionally identify and correct certain inaccuracies present in previous analyses by other researchers. Furthermore, we provide Python-based computations to validate the theoretical findings, accompanied by comparative numerical and graphical illustrations.

References

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