Sombor index of zero-divisor graphs of commutative rings

Gürsoy, Arif; Ülker, Alper; Kircali Gürsoy, Necla

doi:10.2478/auom-2022-0028

Abstract

In this paper, we investigate the Sombor index of the zero-divisor graph of ℤ_n which is denoted by Γ(ℤ_n) for n ∈ {p^α, pq, p²q, pqr} where p, q and r are distinct prime numbers. Moreover, we introduce an algorithm which calculates the Sombor index of Γ(ℤ_n). Finally, we give Sombor index of product of rings of integers modulo n.

References

[1] M. Aijaz and S. Pirzada, Annihilating-ideal graphs of commutative rings, Asian-European Journal of Mathematics, 13(2020).10.1142/S1793557120501211
Search in Google Scholar Back to article
[2] S. Akbari, D. Kiani, F. Mohammadi, and S. Moradi, The total graph and regular graph of a commutative ring, Journal of Pure and Applied Algebra, 213(2009).10.1016/j.jpaa.2009.03.013
Search in Google Scholar Back to article
[3] N. Akgunes and Y. Nacaroglu, Some properties of zero divisor graph obtained by the ring z p z q z r, Asian-European Journal of Mathematics, 12(2019).10.1142/S179355712040001X
Search in Google Scholar Back to article
[4] N. Akgunes and M. Togan, Some graph theoretical properties over zero-divisor graphs of special finite commutative rings, Advanced Studies in Contemporary Mathematics (Kyungshang), 22(2012).
Search in Google Scholar Back to article
[5] A. Ali, K. C. Das, and S. Akhter, On the extremal graphs for second Zagreb index with fixed number of vertices and cyclomatic number, Miskolc Math. Notes (2019).
Search in Google Scholar Back to article
[6] S. Alikhani and N. Ghanbari, Sombor index of polymers, Match, 86(2021).
Search in Google Scholar Back to article
[7] M. An and K. C. Das, First Zagreb index, k-connectivity, beta-deficiency and k-hamiltonicity of graphs, MATCH Commun. Math. Comput. Chem, 80(2018), 141–151.
Search in Google Scholar Back to article
[8] D. F. Anderson and A. Badawi, The total graph of a commutative ring, Journal of Algebra, 320(2008).10.1016/j.jalgebra.2008.06.028
Search in Google Scholar Back to article
[9] D. F. Anderson and P. S. Livingston, The zero-divisor graph of a commutative ring, Journal of algebra, 217(1999), 434–447.10.1006/jabr.1998.7840
Search in Google Scholar Back to article
[10] T. Asir and V. Rabikka, The Wiener index of the zero-divisor graph of zn, Discrete Applied Mathematics (2021).
Search in Google Scholar Back to article
[11] I. Beck, Coloring of commutative rings, Journal of Algebra, 116(1988).10.1016/0021-8693(88)90202-5
Search in Google Scholar Back to article
[12] B. Borovicanin, K. C. Das, B. Furtula, and I. Gutman, Zagreb indices: Bounds and extremal graphs, Bounds in Chemical Graph TheoryBasics, Univ. Kragujevac, Kragujevac, 67(153)(2017).
Search in Google Scholar Back to article
[13] L. Buyantogtokh, B. Horoldagva, and K. C. Das, On reduced second Zagreb index, Journal of Combinatorial Optimization, 39(2020).10.1007/s10878-019-00518-7
Search in Google Scholar Back to article
[14] R. Cruz, I. Gutman, and J. Rada, Sombor index of chemical graphs, Applied Mathematics and Computation, 399(2021).10.1016/j.amc.2021.126018
Search in Google Scholar Back to article
[15] K. C. Das and A. Ali, On a conjecture about the second Zagreb index, Discrete Mathematics Letters, 2(2019).
Search in Google Scholar Back to article
[16] K. C. Das, A. S. evik, I. N. Cangul, and Y. Shang, On Sombor index, Symmetry, 13(2021).10.3390/sym13010140
Search in Google Scholar Back to article
[17] H. Deng, Z. Tang, and R. Wu, Molecular trees with extremal values of Sombor indices, International Journal of Quantum Chemistry, 121(2021).10.1002/qua.26622
Search in Google Scholar Back to article
[18] S. Fajtlowicz, On conjectures of graffiti-ii, Congr. Numer, 60(1987), 187–197.
Search in Google Scholar Back to article
[19] X. Fang, L. You, and H. Liu, The expected values of Sombor indices in random hexagonal chains, phenylene chains and Sombor indices of some chemical graphs, International Journal of Quantum Chemistry, 121(2021).10.1002/qua.26740
Search in Google Scholar Back to article
[20] S. Filipovski, Relations between Sombor index and some degreebased topological indices, Iranian Journal of Mathematical Chemistry, 12(2021).
Search in Google Scholar Back to article
[21] B. Furtula and I. Gutman, A forgotten topological index, Journal of Mathematical Chemistry, 53(2015).10.1007/s10910-015-0480-z
Search in Google Scholar Back to article
[22] N. Ghanbari and S. Alikhani, Sombor index of certain graphs, Iranian Journal of Mathematical Chemistry, 12(2021).
Search in Google Scholar Back to article
[23] I. Gutman, Degree-based topological indices, Croatica chemica acta, 86(2013), 351–361.10.5562/cca2294
Search in Google Scholar Back to article
[24] I. Gutman, Geometric approach to degree-based topological indices: Sombor indices, MATCH Commun. Math. Comput. Chem, 86(2021), 11–16.
Search in Google Scholar Back to article
[25] I. Gutman, Some basic properties of Sombor indices, Open Journal of Discrete Applied Mathematics, 4(2021), 1–3.10.30538/psrp-odam2021.0047
Search in Google Scholar Back to article
[26] I. Gutman and O. E. Polansky, Mathematical Concepts in Organic Chemistry, Springer Science & Business Media, 1986.10.1515/9783112570180
Search in Google Scholar Back to article
[27] I. Gutman and N. Trinajstić, Graph theory and molecular orbitals. total ϕ-electron energy of alternant hydrocarbons, Chemical Physics Letters, 17(1972).10.1016/0009-2614(72)85099-1
Search in Google Scholar Back to article
[28] B. Horoldagva and K. C. Das, On Zagreb indices of graphs, Match, 85(2021).
Search in Google Scholar Back to article
[29] B. Horoldagva, K. C. Das, and T. A. Selenge, Complete characterization of graphs for direct comparing Zagreb indices, Discrete Applied Mathematics, 215(2016).10.1016/j.dam.2016.07.008
Search in Google Scholar Back to article
[30] B. Horoldagva and C. Xu, On Sombor index of graphs, MATCH Commun. Math. Comput. Chem, 86(2021), 703–713.
Search in Google Scholar Back to article
[31] V. R. Kulli and I. Gutman, Computation of Sombor indices of certain networks, International Journal of Applied Chemistry, 8(2021).10.14445/23939133/IJAC-V8I1P101
Search in Google Scholar Back to article
[32] Z. Lin and L. Miao, On the spectral radius, energy and Estrada index of the Sombor matrix of graphs, arXiv preprint arXiv:2102.03960(2021).
Search in Google Scholar Back to article
[33] I. Milovanovic, E. Milovanovic, and M. Matejic, On some mathematical properties of Sombor indices, Bull. Int. Math. Virtual Inst, 11(2021), 341–353.
Search in Google Scholar Back to article
[34] M. Randić, On characterization of molecular branching, Journal of the American Chemical Society, 97(1975).10.1021/ja00856a001
Search in Google Scholar Back to article
[35] I. Redžepović, Chemical applicability of Sombor indices, Journal of the Serbian Chemical Society, 86(2021).10.2298/JSC201215006R
Search in Google Scholar Back to article
[36] T. Réti, T. Došlic, and A. Ali, On the Sombor index of graphs, Contrib. Math, 3(2021), 11–18.
Search in Google Scholar Back to article
[37] S. Salehifar, K. Khashyarmanesh, and M. Afkhami, On the annihilator-ideal graph of commutative rings, Ricerche di Matematica, 66(2017).10.1007/s11587-016-0311-y
Search in Google Scholar Back to article
[38] Y. Shang, On the number of spanning trees, the laplacian eigenvalues, and the laplacian Estrada index of subdivided-line graphs, Open Mathematics, 14(2016).10.1515/math-2016-0055
Search in Google Scholar Back to article
[39] Y. Shang, Lower bounds for Gaussian Estrada index of graphs, Symmetry, 10(2018).10.3390/sym10080325
Search in Google Scholar Back to article
[40] F. Shaveisi, Some results on annihilating-ideal graphs, Canadian Mathematical Bulletin, 59(2016).10.4153/CMB-2016-016-3
Search in Google Scholar Back to article
[41] P. Singh and V. K. Bhat, Adjacency matrix and Wiener index of zero divisor graph γ(ℤn), Journal of Applied Mathematics and Computing, 66(2021).10.1007/s12190-020-01460-2
Search in Google Scholar Back to article
[42] D. Sinha and A. K. Rao, A note on co-maximal graphs of commutative rings, AKCE International Journal of Graphs and Combinatorics, 15(2018).10.1016/j.akcej.2018.03.003
Search in Google Scholar Back to article
[43] A. Tehranian and H. R. Maimani, A study of the total graph, Iranian Journal of Mathematical Sciences and Informatics, 6(2011).
Search in Google Scholar Back to article
[44] R. Todeschini and V. Consonni, Molecular Descriptors for Chemoinformatics, volume 2, John Wiley & Sons, 2010.10.1002/9783527628766
Search in Google Scholar Back to article
[45] H. J. Wang, Graphs associated to co-maximal ideals of commutative rings, Journal of Algebra, 320(2008).10.1016/j.jalgebra.2008.06.020
Search in Google Scholar Back to article
[46] Z. Wang, Y. Mao, Y. Li, and B. Furtula, On relations between Sombor and other degree-based indices, Journal of Applied Mathematics and Computing (2021).10.1007/s12190-021-01516-x
Search in Google Scholar Back to article
[47] H. Wiener, Structural determination of paraffin boiling points, Journal of the American Chemical Society, 69(1947).10.1021/ja01193a00520291038
Search in Google Scholar Back to article
[48] K. Xu, F. Gao, K. C. Das, and N. Trinajstić, A formula with its applications on the difference of Zagreb indices of graphs, Journal of Mathematical Chemistry, 57(2019).10.1007/s10910-019-01025-0
Search in Google Scholar Back to article
[49] M. Ye and T. Wu, Co-maximal ideal graphs of commutative rings, Journal of Algebra and its Applications, 11(2012).10.1142/S0219498812501149
Search in Google Scholar Back to article
[50] M. Young, Adjacency matrices of zero-divisor graphs of integers modulo n, Involve, a Journal of Mathematics, 8(2015).10.2140/involve.2015.8.753
Search in Google Scholar Back to article
[51] T. Zhou, Z. Lin, and L. Miao, The Sombor index of trees and unicyclic graphs with given matching number, arXiv preprint arXiv:2103.04645(2021).10.1080/09720529.2021.2015090
Search in Google Scholar Back to article

Sombor index of zero-divisor graphs of commutative rings

Abstract

Paradigm

My account