References
- [1] M. Aijaz and S. Pirzada, Annihilating-ideal graphs of commutative rings, Asian-European Journal of Mathematics, 13(2020).10.1142/S1793557120501211
- [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
- [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
- [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).
- [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).
- [6] S. Alikhani and N. Ghanbari, Sombor index of polymers, Match, 86(2021).
- [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.
- [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
- [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
- [10] T. Asir and V. Rabikka, The Wiener index of the zero-divisor graph of zn, Discrete Applied Mathematics (2021).
- [11] I. Beck, Coloring of commutative rings, Journal of Algebra, 116(1988).10.1016/0021-8693(88)90202-5
- [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).
- [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
- [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
- [15] K. C. Das and A. Ali, On a conjecture about the second Zagreb index, Discrete Mathematics Letters, 2(2019).
- [16] K. C. Das, A. S. evik, I. N. Cangul, and Y. Shang, On Sombor index, Symmetry, 13(2021).10.3390/sym13010140
- [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
- [18] S. Fajtlowicz, On conjectures of graffiti-ii, Congr. Numer, 60(1987), 187–197.
- [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
- [20] S. Filipovski, Relations between Sombor index and some degreebased topological indices, Iranian Journal of Mathematical Chemistry, 12(2021).
- [21] B. Furtula and I. Gutman, A forgotten topological index, Journal of Mathematical Chemistry, 53(2015).10.1007/s10910-015-0480-z
- [22] N. Ghanbari and S. Alikhani, Sombor index of certain graphs, Iranian Journal of Mathematical Chemistry, 12(2021).
- [23] I. Gutman, Degree-based topological indices, Croatica chemica acta, 86(2013), 351–361.10.5562/cca2294
- [24] I. Gutman, Geometric approach to degree-based topological indices: Sombor indices, MATCH Commun. Math. Comput. Chem, 86(2021), 11–16.
- [25] I. Gutman, Some basic properties of Sombor indices, Open Journal of Discrete Applied Mathematics, 4(2021), 1–3.10.30538/psrp-odam2021.0047
- [26] I. Gutman and O. E. Polansky, Mathematical Concepts in Organic Chemistry, Springer Science & Business Media, 1986.10.1515/9783112570180
- [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
- [28] B. Horoldagva and K. C. Das, On Zagreb indices of graphs, Match, 85(2021).
- [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
- [30] B. Horoldagva and C. Xu, On Sombor index of graphs, MATCH Commun. Math. Comput. Chem, 86(2021), 703–713.
- [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
- [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).
- [33] I. Milovanovic, E. Milovanovic, and M. Matejic, On some mathematical properties of Sombor indices, Bull. Int. Math. Virtual Inst, 11(2021), 341–353.
- [34] M. Randić, On characterization of molecular branching, Journal of the American Chemical Society, 97(1975).10.1021/ja00856a001
- [35] I. Redžepović, Chemical applicability of Sombor indices, Journal of the Serbian Chemical Society, 86(2021).10.2298/JSC201215006R
- [36] T. Réti, T. Došlic, and A. Ali, On the Sombor index of graphs, Contrib. Math, 3(2021), 11–18.
- [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
- [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
- [39] Y. Shang, Lower bounds for Gaussian Estrada index of graphs, Symmetry, 10(2018).10.3390/sym10080325
- [40] F. Shaveisi, Some results on annihilating-ideal graphs, Canadian Mathematical Bulletin, 59(2016).10.4153/CMB-2016-016-3
- [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
- [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
- [43] A. Tehranian and H. R. Maimani, A study of the total graph, Iranian Journal of Mathematical Sciences and Informatics, 6(2011).
- [44] R. Todeschini and V. Consonni, Molecular Descriptors for Chemoinformatics, volume 2, John Wiley & Sons, 2010.10.1002/9783527628766
- [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
- [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
- [47] H. Wiener, Structural determination of paraffin boiling points, Journal of the American Chemical Society, 69(1947).10.1021/ja01193a00520291038
- [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
- [49] M. Ye and T. Wu, Co-maximal ideal graphs of commutative rings, Journal of Algebra and its Applications, 11(2012).10.1142/S0219498812501149
- [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
- [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