Metric dimension and Zagreb indices of essential ideal graph of a finite commutative ring

Document Type : Research Paper

Authors

1 PG and Research Department of Mathematics, Farook College, P.O. Farook College, Kozhikode, Kerala, India- 673632

2 Department of Mathematics, National Institute of Technology Calicut, Kozhikode, 673601, Kerala, India

Abstract

Let $R$ be a commutative ring with unity. The essential ideal graph $\mathcal{E}_{R}$ of $R$ is a graph whose set of vertex consists of all nonzero proper ideals of R. Two vertices $\hat{I}$ and $\hat{J}$ are adjacent if and only if $\hat{I}+ \hat{J}$ is an essential ideal. In this paper, we characterize the graph $\mathcal{E}_{R}$ as having a finite metric dimension. Furthermore, we identify that the essential ideal graph and the annihilating ideal graph of the ring $\mathbb{Z}_{n}$ are isomorphic whenever $n$ is a product of distinct primes. In addition, we estimate the metric dimension of the essential ideal graph of the ring $\mathbb{Z}_{n}$. Moreover, we determine the topological indices, namely the first and second Zagreb indices, of $\mathcal{E}_{\mathbb Z_n}$.

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References

[1] M. R. Ahmadi and R. Jahani-Nezhad, Energy and Wiener index of zero-divisor graphs, Iranian J. Math. Chem., 2 no. 1 (2011) 45–51.
[2] M. Aijaz and S. Pirzada, Annihilating-ideal graphs of commutative rings, Asian-Eur. J. Math. , 13 no. 7 (2020) 12 pp.
[3] J. Amjadi, The essential ideal graph of a commutative ring, Asian-Eur. J. Math., 11 no. 4 (2018) 9 pp.
[4] S. Banerjee, Spectra and topological indices of comaximal graph of Zn, Results Math., 77 no. 3 (2022) 23 pp.
[5] S. Banerjee, The adjacency spectrum and metric dimension of an induced subgraph of comaximal graph of Zn, Discrete Math. Algorithms Appl., 15 no. 3 (2023) 17 pp.
[6] M. Behboodi and Z. Rakeei, The annihilating-ideal graph of commutative rings I, J. Algebra Appl., 10 no. 4 (2011) 727–739.
[7] G. Chartrand, L. Eroh, M. A. Johnson and O. R. Oellermann, Resolvability in graphs and the metric dimension of a graph, Discrete Appl. Math., 105 no. 1-3 (2000) 99–113.
[8] V. Consonni and R. Todeschini, Molecular descriptors for chemoinformatics: volume I: alphabetical listing/volume II: appendices, references, John Wiley & Sons, 2009.
[9] B. Furtula and I. Gutman, A forgotten topological index, J. Math. Chem., 53 no. 4 (2015) 1184–1190.
[10] I. Gutman, Geometric approach to degree-based topological indices: Sombor indices, MATCH Commun. Math. Comput. Chem., 86 no. 1 (2021) 11–16.
[11] I. Gutman and O. E. Polansky, Mathematical concepts in organic chemistry, Springer-Verlag, Berlin, 1986.
[12] I. Gutman and N. Trinajsti´c, Graph theory and molecular orbitals. Total ϕ-electron energy of alternant hydrocarbons, Chem. Phys. Lett., 17 no. 4 (1972) 535–538.
[13] I. Gutman, B. Ru˘s˘ci´c, N. Trinajsti´c and C. F. Wilcox, Graph theory and molecular orbitals. XII. Acyclic polyenes, J. Chem. Phys., 62 no. 9 (1975) 3399–3405.
[14] F. Harary and R. A. Melter, On the metric dimension of a graph, Ars combin., 2 (1976) 191–195.
[15] B. L. Hulme, A. W. Shiver and P. J. Slater, A Boolean algebraic analysis of fire protection, North-Holland Math. Stud., 95, Ann. Discrete Math., 19, North-Holland, Amsterdam, 1984.
[16] P. Jamsheena and A. V. Chithra, Adjacency Spectrum and Wiener Index of the Essential Ideal Graph of a Finite Commutative Ring Zn, (2023) In: arXiv preprint arXiv:2303.08468.
[17] P. Jamsheena and A. V. Chithra, On the structure and spectra of an induced subgraph of essential ideal graph of Zn, (2023) In: arXiv preprint arXiv:2310.10999.
[18] M. Johnson, Structure-activity maps for visualizing the graph variables arising in drug design, J. Biopharm. Stat., 3 no. 2 (1993) 203–236.
[19] M. A. Johnson, Browsable structure-activity datasets, Adv. Mol. Similarity, 2 (1998) 153– 170.
[20] S. Khuller, B. Raghavachari and A. Rosenfeld, Landmarks in graphs, Discrete Appl. Math., 70 no. 3 (1996) 217–229.
[21] R. A. Melter and I. Tomescu, Metric bases in digital geometry, Computer vision, graphics, and image Processing, 25 no. 1 (1984) 113–121.
[22] S. Pirzada, R. Raja and S. Redmond, Locating sets and numbers of graphs associated to commutative rings, J. Algebra Appl., 13 no. 7 (2014) 18 pp.
[23] R. Raja, S. Pirzada and S. Redmond, On locating numbers and codes of zero divisor graphs associated with commutative rings, J. Algebra Appl., 15 no. 1 (2016) 22 pp.
[24] K. Selvakumar, P. Gangaeswari and G. Arunkumar, The Wiener index of the zero-divisor graph of a finite commutative ring with unity, Discrete Appl. Math., 311 (2022) 72–84.
[25] P. J. Slater, Leaves of trees, Proceedings of the Sixth Southeastern Conference on Combinatorics, Graph Theory and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1975), Congr. Numer, (1975) 549–559.

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History

  • Receive Date: 07 June 2024
  • Revise Date: 22 March 2025
  • Accept Date: 19 May 2025
  • Published Online: 26 July 2025

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