Abstract
In this paper, we investigate the first and second Zagreb multiplicative indices of zero divisor graphs of reduced rings from an applied perspective. The zero-divisor graph of a ring, denoted by \(\Gamma (R)\), consists of non-zero zero-divisors of a ring R as its vertex set, with two vertices connected by an edge if their product is zero. Recently, in Selvakumar et al. (Discr Appl Math 311:72–84, 2022), we explored the Wiener index of the zero divisor graph under various conditions: (i) when R is a reduced ring, (ii) when R is the ring of integers modulo n, and (iii) more generally when R is the product of the rings of integers modulo n. This paper extends that work by examining the Zagreb multiplicative indices, another significant topological index, utilizing analytical methods. We provide explicit formulas for these indices specifically when the ring R is reduced. The applicability of these formulas is demonstrated through numerous examples provided.
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References
Alvaro, M.P., and R.M. José. 2022. New lower bounds for the first variable Zagreb index. Discrete Applied Mathematics 306: 166–173.
Anderson, D.F., and P.S. Livingston. 1999. The zero-divisor graph of a commutative ring. Journal of Algebra 217: 434–447.
Gutman, I., and N. Trinajstic. 1972. Graph theory, and molecular orbitals, total \(\pi\)-electron energy of alternant hydrocarbons. Chemical Physics Letters 17: 535–538.
Mehdi, E. 2022. Unicyclic and bicyclic graphs with maximum exponential second Zagreb index. Discrete Applied Mathematics 307: 172–179.
Muhuo, L., C. Kun, and F. Boris. 2021. Minimum augmented Zagreb index of c-cyclic graphs. Discrete Applied Mathematics 295: 32–38.
Schwenk, A.J. 1974. Computing the characteristic polynomial of a graph. In Graphs combinatorics, vol. 406, ed. R. Bary and F. Harary. Lecture Notes in Mathematics, 153–172. Berlin: Springer-Verlag.
Selvakumar, K., B. Beautlin Jemi, and A.R. Moniri Hamzekolaee. 2024. A topological property of a hypergraph assigned to commutative rings. Journal of Algebra and Its Applications. https://doi.org/10.1142/S0219498825500938. Article in Press.
Selvakumar, K., P. Gangaeswari, and G. Arunkumar. 2022. The Wiener index of the zero-divisor graph of a finite commutative ring with unity. Discrete Applied Mathematics 311: 72–84.
Singh, P., and V.K. Bhat. 2020. Zero-divisor graphs of finite commutative rings: a survey, surveys in math. Discrete Applied Mathematics 15: 371–397.
Todeschini, R., and V. Consonni. 2000. Handbook of molecular descriptors. Weinheim: Wiley-VCH.
West, D.B. 2001. Introduction to graph theory, 2nd ed. Singapore: Pearson Education.
Wiener, H. 1947. Structural determination of paraffin boiling points. Journal of the American Chemical Society 69 (1): 17–20.
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Communicated by S. Ponnusamy
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The corresponding author and she acknowledges the CSIR Junior Research Fellowship (09/0652(11961)/2021-EMR-I).
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Selvakumar, K., Gangaeswari, P. Some applications of multiplicative Zagreb index. J Anal (2024). https://doi.org/10.1007/s41478-024-00771-y
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DOI: https://doi.org/10.1007/s41478-024-00771-y