The Wiener Index of the Associate Graph of $Z_n$

Articles in Press

Document Type : Original paper

Authors

1 Department of Mathematics, Hemchandracharya North Gujarat University, Patan - 384265, Gujarat, India

2 Research Scholar, Department of Mathematics, Hemchandracharya North Gujarat University, Patan - 384265, Gujarat, India

Abstract

Let $R$ be a commutative ring, the associate graph, $Ass(R)$ of ring $R$ has the elements of ring $R$ as vertices and two distinct vertices $u$ and $v$ are adjacent if $u$ and $v$ are associate elements of $R$. In this article we investigate the Wiener index of $Ass(\mathbb{Z}_{n})$ and its line graph for all $n\in \mathbb{N}$. We also give some characterization results regarding degree, diameter, girth, clique number, chromatic number, domination number, and independence number of $Ass(\mathbb{Z}_{n})$ and $L\left( Ass(\mathbb{Z}_{n}) \right)$.

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References

[1] T. Asir and V. Rabikka, The Wiener index of the zero-divisor graph of Zn, Discrete Appl. Math. 319 (2022), no. 38, 461–471. https://doi.org/10.1016/j.dam.2021.02.035
[2] I. Beck, Coloring of commutative rings, J. Algebra 116 (1988), no. 1, 208–226. https://doi.org/10.1016/0021–8693(88)90202-5
[3] L. Boro, M.M. Singh, and J. Goswami, Line graph associated to the intersection graph of ideals of rings, J. Math. Comput. Sci. 11 (2021), no. 3, 2736–2745. https://doi.org/10.28919/jmcs/5567
[4] L. Boro, M.M. Singh, and J. Goswami, On the line graphs associated to the unit graphs of rings, Palestine J. Math. 11 (2022), no. 4, 139–145.
[5] D.M. Burton, Elementary Number Theory, McGraw-Hill Companies, 2011.
[6] H.J. Chiang-Hsieh, P.F. Lee, and H.J. Wang, The embedding of line graphs associated to the zero-divisor graphs of commutative rings, Israel J. Math. 180 (2010), no. 1, 193–222. https://doi.org/10.1007/s11856-010-0101-2
[7] J. Clark and D.A. Holton, A First Look at Graph Theory, Allied Publishers Ltd., 1995.
[8] M. Dehmer and F. Emmert-Streib, Quantitative Graph Theory: Mathematical Foundations and Applications, CRC Press, 2014.
[9] J. Devillers and A. Balaban, Topological indices and related descriptors in QSAR and QSPR, Gordon and Breach Science Publishers, 1999.
[10] A. Dobrynin, R. Entringer, and I. Gutman, Wiener index of trees: Theory and applications, Acta Appl. Math. 66 (2001), no. 3, 211–249. https://doi.org/10.1023/A:1010767517079
[11] A. Dobrynin and A. Iranmanesh, Wiener index of edge thorny graphs of catacondensed benzenoids, Mathematics 8 (2020), no. 4, 467–479. https://doi.org/10.3390/math8040467
[12] K. Elahi, A. Ahmad, and R. Hasni, Construction algorithm for zero divisor graphs of finite commutative rings and their vertex-based eccentric topological indices, Mathematics 6 (2018), no. 12, 301–309. https://doi.org/10.3390/math6120301
[13] N. Gohain, T. Ali, and A. Akhtar, Reducing redundancy of codons through total graph, Current Research in Bioinformatics 4 (2015), no. 1, 1–6. https://doi.org/10.3844/ajbsp.2015.1.6
[14] J. Goswami and H.K. Saikia, On the line graph associated to the line graph of a module, Matematika 31 (2015), no. 1, 7–13. https://doi.org/10.11113/matematika.v31.n1.676
[15] I. Gutman and O.E. Polansky, Mathematical Concept in Organic Chemistry, Springer-Verlag, 1986.
[16] F. Harary, Graph Theory, Addison-Wesley Publishing Company, 1969.
[17] F. Harary and R.Z. Norman, Some properties of line digraphs, Rendiconti del Circolo Matematico di Palermo 9 (1960), no. 2, 161–169. https://doi.org/10.1007/BF02854581
[18] T. Haynes, S.T. Hedetniemi, and P.J. Slater, Fundamentals of Domination in Graphs, Marcel Dekker, New York, 1988.
[19] I.N. Herstein, Topics in Algebra, John Wiley & Sons, 1975.
[20] H. Hosaya, Topological index. a newly proposed quantity characterizing the topological nature of structural isomers of saturated hydrocarbons, Bull. Chem. Soc. Japan 44 (1971), no. 9, 2332–2339. https://doi.org/10.1246/bcsj.44.2332
[21] N.J. Khalel and N.E. Arif, Associate graph of a commutative ring, J. Discrete Math. Sci. Cryp. 26 (2023), no. 7, 1883–1887. https://doi.org/10.47974/JDMSC-1679
[22] M. Sarmah and K. Patra, Line graph associated to total graph of idealization, Afrika Matematika 27 (2015), no. 3, 485–490. https://doi.org/10.1007/s13370-015-0355-2
[23] M.J. Subhakar, Associate ring graphs, Mapana J. Sciences 9 (2010), no. 1, 31–40. https://doi.org/10.12723/mjs.16.5
[24] C. Susanth and S.J. Kalayathankal, The sum and product of independence numbers of graphs and their line graphs, J. Inf. Math. Sci. 6 (2014), no. 2, 77–85.
[25] H. Wiener, Structural determination of paraffin boiling points, J. American Chem. Soc. 69 (1947), no. 1, 17–20. https://doi.org/10.1021/ja01193a005
[26] Z.P. Zoran and S.P. Zoran, The line graph associated to the total graph of commutative rings, Ars Combin. 127 (2016), 185–195.
Communications in Combinatorics and Optimization

Articles in Press, Accepted Manuscript
Available Online from 16 April 2026

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