On properties and topological indices of the zero divisor graph for direct product of some commutative rings

Nurhabibah; Abdul Gazir Syarifudin; Intan Muchtadi-Alamsyah; Erma Suwastika; Nor Haniza Sarmin; Nur Idayu Alimon; Ghazali Semil @ Ismail; Nurhabibah; Abdul Gazir Syarifudin; Intan Muchtadi-Alamsyah; Erma Suwastika; Nor Haniza Sarmin; Nur Idayu Alimon; Ghazali Semil @ Ismail

doi:10.3934/math.2026066

AIMS Mathematics

2026, Volume 11, Issue 1: 1590-1615. doi: 10.3934/math.2026066

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Research article

On properties and topological indices of the zero divisor graph for direct product of some commutative rings

1.
Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung, Bandung 40132, Indonesia
2.
Faculty of Mathematics and Natural Sciences, Universitas Kebangsaan Republik Indonesia, Bandung 40263, Indonesia
3.
Department of Mathematical Sciences, Faculty of Science, Universiti Teknologi Malaysia, Johor Bahru 81310, Johor, Malaysia
4.
Faculty of Computer and Mathematical Sciences, Universiti Teknologi MARA Johor Branch, Pasir Gudang Campus, Masai, Johor 81750, Malaysia

Received: 09 November 2025 Revised: 26 December 2025 Accepted: 08 January 2026 Published: 19 January 2026
MSC : 05C07, 05C09, 05C25, 13A70

Graph-theoretic representations of algebraic structures provide useful tools for studying algebraic properties and degree-distance-based invariants that arise, for example, in mathematical chemistry. Motivated by these applications and by recent work on zero divisor graphs over modular rings, this paper investigated the zero divisor graph $ \Gamma(R) $ of a commutative ring $ R $ in the sense of Anderson and Livingston, where vertices are the zero divisors and two distinct vertices are adjacent when their product is zero. We focused on rings of the form $ \mathbb{Z}_{p^m} \times \mathbb{Z}_{q^n} $, with $ p, q $ primes and $ m, n \in \mathbb{N} $. This research employed a literature review combined with a deductive method, drawing specific conclusions from well-established algebraic and graph-theoretic principles. By explicitly describing the zero divisors of $ R $ and partitioning the vertex set according to greatest common divisor conditions, we determined several structural properties of $ \Gamma(\mathbb{Z}_{p^m} \times \mathbb{Z}_{q^n}) $. Furthermore, we obtained closed formulas for the general zeroth-order Randić index, the eccentric connectivity index, and the Schultz index of this graph. These results extended previous studies on zero divisor graphs over integers modulo rings and yield explicit families of graphs with exactly computable topological indices.
- zero divisor graph,
- integers modulo ring,
- graph properties,
- topological indices
Citation: Nurhabibah, Abdul Gazir Syarifudin, Intan Muchtadi-Alamsyah, Erma Suwastika, Nor Haniza Sarmin, Nur Idayu Alimon, Ghazali Semil @ Ismail. On properties and topological indices of the zero divisor graph for direct product of some commutative rings[J]. AIMS Mathematics, 2026, 11(1): 1590-1615. doi: 10.3934/math.2026066

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Abstract

Graph-theoretic representations of algebraic structures provide useful tools for studying algebraic properties and degree-distance-based invariants that arise, for example, in mathematical chemistry. Motivated by these applications and by recent work on zero divisor graphs over modular rings, this paper investigated the zero divisor graph $ \Gamma(R) $ of a commutative ring $ R $ in the sense of Anderson and Livingston, where vertices are the zero divisors and two distinct vertices are adjacent when their product is zero. We focused on rings of the form $ \mathbb{Z}_{p^m} \times \mathbb{Z}_{q^n} $, with $ p, q $ primes and $ m, n \in \mathbb{N} $. This research employed a literature review combined with a deductive method, drawing specific conclusions from well-established algebraic and graph-theoretic principles. By explicitly describing the zero divisors of $ R $ and partitioning the vertex set according to greatest common divisor conditions, we determined several structural properties of $ \Gamma(\mathbb{Z}_{p^m} \times \mathbb{Z}_{q^n}) $. Furthermore, we obtained closed formulas for the general zeroth-order Randić index, the eccentric connectivity index, and the Schultz index of this graph. These results extended previous studies on zero divisor graphs over integers modulo rings and yield explicit families of graphs with exactly computable topological indices.

References

[1]	H. Wiener, Structural determination of paraffin boiling points, J. Amer. Chem. Soc., 69 (1947), 17–20. https://doi.org/10.1021/ja01193a005 doi: 10.1021/ja01193a005
[2]	H. Hosoya, Topological index. A newly proposed quantity characterizing the topological nature of structural isomers of saturated hydrocarbons, Bull. Chem. Soc. Jpn., 44 (1971), 2332–2339. https://doi.org/10.1246/bcsj.44.2332 doi: 10.1246/bcsj.44.2332
[3]	M. Randić, On characterization of molecular branching, J. Amer. Chem. Soc., 97 (1975), 6609–6615. https://doi.org/10.1021/ja00856a001 doi: 10.1021/ja00856a001
[4]	H. P. Schultz, Topological organic chemistry. 1. Graph theory and topological indices of alkane, J. Chem. Inf. Comput. Sci., 29 (1989), 227–228. https://doi.org/10.1021/ci00063a012 doi: 10.1021/ci00063a012
[5]	X. Li, J. Zheng, A unified approach to the extremal trees for different indices, MATCH Commun. Math. Comput. Chem., 54 (2005), 195–208. https://doi.org/10.1111/j.1467-9892.2005.00394.x doi: 10.1111/j.1467-9892.2005.00394.x
[6]	V. Sharma, R. Goswami, A. K. Madan, Eccentric connectivity index: a novel highly discriminating topological descriptor for structure-property and structure-activity studies, J. Chem. Inf. Comput. Sci., 37 (1997), 273–282. https://doi.org/10.1021/ci960049h doi: 10.1021/ci960049h
[7]	I. Beck, Coloring of commutative ring, J. Algebra, 116 (1988), 208–226. https://doi.org/10.1016/0021-8693(88)90202-5 doi: 10.1016/0021-8693(88)90202-5
[8]	D. F. Anderson, P. S. Livingston, The zero divisor-graph of a commutative ring, J. Algebra, 217 (1999), 434–447. https://doi.org/10.1006/jabr.1998.7840 doi: 10.1006/jabr.1998.7840
[9]	F. DeMeyer, K. Schneider, Automorphisms and zero-divisor graphs of commutative rings, Int. J. Commut. Rings, 1 (2002), 93–106.
[10]	S. B. Mulay, Cycles and symmetries of zero-divisors, Commun. Algebra, 30 (2002), 3533–3558. https://doi.org/10.1081/AGB-120004502 doi: 10.1081/AGB-120004502
[11]	S. P. Redmond, The zero-divisor graph of a non-commutative ring, Int. J. Commut. Rings, 1 (2002), 203–211.
[12]	S. P. Redmond, An ideal-based zero-divisor graph of a commutative ring, Commun. Algebra, 31 (2003), 4425–4443. https://doi.org/10.1081/AGB-120022801 doi: 10.1081/AGB-120022801
[13]	M. Axtell, J. Coykendall, J. Stickles, Zero-divisor graphs of polynomial and power series over commutative rings, Commun. Algebra, 33 (2005), 2043–2050. https://doi.org/10.1081/AGB-200063357 doi: 10.1081/AGB-200063357
[14]	M. Axtell, J. Stickles, J. Warfel, Zero-divisor graphs of direct products of commutative rings, Houston J. Math., 32 (2006), 985–994.
[15]	A. Phillips, J. Rogers, K. Tolliver, F. Worek, Uncharted territory of zero-divisor graphs and their complements, Sumsri, 2004.
[16]	N. Cordova, C. Gholston, H. Hauser, The structure of zero-divisor graphs, Sumsri, 2005.
[17]	C. I. Aponte, P. S. Johnson, N. A. Mims, Line graphs of zero-divisor graphs, Sumsri, 2005.
[18]	L. M. Birch, J. J. Thibodeaux, R. P. Tucci, Zero-divisor graphs of finite direct products of finite rings, Commun. Algebra, 42 (2014), 3852–3860. https://doi.org/10.1080/00927872.2013.796556 doi: 10.1080/00927872.2013.796556
[19]	A. N. Koam, A. Ahmad, A. Haider, On eccentric topological indices based on edges of zero-divisor graphs, Symmetry, 11 (2019), 907. https://doi.org/10.3390/sym11070907 doi: 10.3390/sym11070907
[20]	N. Akgüneş, Y. Nacaroğlu, Some properties of zero-divisor graph obtained by the ring $\mathbb{Z}_p \times \mathbb{Z}_q \times \mathbb{Z}_r$, Asian Eur. J. Math., 12 (2019), 2040001. https://doi.org/10.1142/S179355712040001X doi: 10.1142/S179355712040001X
[21]	P. Singh, V. K. Bhat, Zero-divisor graphs of finite commutative rings: a survey, Surv. Math. Appl., 15 (2020), 371–397.
[22]	F. Maulana, M. Z. Aditya, E. Suwastika, I. Muchtadi-Alamsyah, N. I. Alimon, N. H. Sarmin, On the topological indices of zero divisor graphs of some commutative rings, J. Appl. Math. Inf., 42 (2024), 663–680. https://doi.org/10.14317/jami.2024.663 doi: 10.14317/jami.2024.663
[23]	G. S. Ismail, N. H. Sarmin, N. I. Alimon, F. Maulana, The first zagreb index of the zero divisor graph for the ring of integers modulo power of primes, Malays. J. Fundam. Appl. Sci., 19 (2023), 892–200. https://doi.org/10.11113/mjfas.v19n5.2980 doi: 10.11113/mjfas.v19n5.2980
[24]	G. S. Ismail, N. H. Sarmin, N. I. Alimon, F. Maulana, General zeroth-order randić index of zero divisor graph for the ring of integers modulo $p^n$, AIP Conf. Proc., 2975 (2023), 020002. https://doi.org/10.1063/5.0181017 doi: 10.1063/5.0181017
[25]	G. Ismail, N. H. Sarmin, N. I. Alimon, F. Maulana, General zeroth-order randić index of the zero divisor graph for some commutative rings, AIP Conf. Proc., 2905 (2024), 070005. https://doi.org/10.1063/5.0171669 doi: 10.1063/5.0171669
[26]	G. S. Ismail, N. H. Sarmin, N. I. Alimon, F. Maulana, The first Zagreb index of the zero divisor graph for the ring of integers modulo $2^kq$, AIP Conf. Proc., 3189 (2024), 110014. https://doi.org/10.1063/5.0225001 doi: 10.1063/5.0225001
[27]	S. Roman, Advanced linear algebra, Springer, 2007. https://doi.org/10.1007/978-0-387-72831-5
[28]	R. Diestel, Graph theory, 5 Eds., Springer, 2017.
[29]	G. H. Hardy, E. M. Wright, An introduction to the theory of numbers, Clarendon Press, 1960. https://doi.org/10.1093/oso/9780199219858.001.0001
[30]	D. F. Anderson, T. Asir, A. Badawi, T. T. Chelvam, Graph of rings, Springer, 2021. https://doi.org/10.1007/978-3-030-88410-9

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