Some of the graph energies of zero-divisor graphs of finite commutative rings

Document Type : Research Paper


Department of Mathematics, Faculty of Sciences, Golestan University, Gorgan, Iran


 In this paper, we investigate some of the graph energies of the zero-divisor graph $\Gamma(R)$ of finite commutative rings $R$. Let $Z(R)$ be the set of zero-divisors of a commutative ring $R$ with non-zero identity and $Z^*(R)=Z(R)\setminus \{0\}$. The zero-divisor graph of $R$, denoted by $\Gamma(R)$, is a simple graph whose vertex set in $Z^*(R)$ and two vertices $u$ and $v$ are adjacent if and only if $uv=vu=0$. We investigate some energies of $\Gamma(R)$ for the commutative rings $R\simeq \mathbb{Z}_{p^2}\times \mathbb{Z}_{q}$, $R\simeq \mathbb{Z}_{p}\times \mathbb{Z}_{p}\times \mathbb{Z}_{p}$ and $R\simeq \mathbb{Z}_{p}\times \mathbb{Z}_{p}\times \mathbb{Z}_{p}\times \mathbb{Z}_{p}$ where $p, q$ the prime numbers.


[1] N. Abreu, D. M. Cardoso, I. Gutman, E. A. Martins and M. Robbiano, Bounds for the signless Laplacian energy, Linear Alg. Appl. 435 (2011), no. 10, 2365–2374.
[2] A. Ahmad, R. Hasni, N. Akhter and K. Elahi, Analysis of distance-based topological polynomials associated with zero-divisor graphs, Comput. Mater. Continua Henderson 70 (2022), no. 2, 2895–2904.
[3] M. H. Akhbari, K. K. Choong and F. Movahedi, A note on the minimum edge dominating energy of graphs, J. Appl. Math. Comput. 63 (2020), 295–310.
[4] D. F. Anderson and P. S. Livingston, The zero-divisor graph of a commutative ring, J. Algebra 167 (1999), 434–447.
[5] S. Arumugam and S. Velammal, Edge Domination in Graphs, Taiwanese J. Math. 2 (1998), no. 2, 173–179.
[6] S. B. Bozkurt and D. Bozkurt, On Incidence Energy, MATCH Commun. Math. Comput. Chem. 72 (2014), 215–225.
[7] S. Chokani, F. Movahedi and S.M. Taheri, Results on some energies of the Zero-divisor graph of the commutative ring Zn, 14th Iran. Int. Group Theory Conf., 2022, pp. 138–146.
[8] D. Cvetkovic, M. Doob and H. Sachs, Spectra of Graphs, Theory and Application, Academic Press, New York, 1980.
[9] K.C. Das and S.A. Mojallal, On Laplacian energy of graphs, Discrete Math. 325 (2014), 52–64.
[10] K.C. Das and S.A. Mojallal, Relation between signless Laplacian energy of graph and its line graph, Linear Alg. Appl. 493 (2016), 91–107.
[11] K.C. Das, S.A. Mojallal and I. Gutman, On energy of line graphs, Linear Alg. Appl. 499 (2016), 79–89.
[12] R.P. Gupta, Independence and covering numbers of line graphs and total graphs. In: Harary, F. (ed.) Proof Techniques in Graph Theory, pp. 61-62, Academic Press, New York, 1969.
[13] I. Gutman, The energy of a graph, Ber. Math-Statist. Sekt. Forschungsz. Graz 103 (1978), 1–22.
[14] I. Gutman and B. Zhou, Laplacian energy of a graph, Linear Alg. Appl. 414 (2006), 29–37.
[15] I. Gutman, M. Robbiano, E.A. Martins, D.M. Cardoso, L. Medina and O. Rojo, Energy of line graphs, Linear Alg. Appl. 433 (2010), 1312–1323.
[16] I. Gutman, N. Trinajstic, Graph theory and molecular orbitals. Total π-electron energy of alternant hydrocarbons, Chem. Phys. Lett. 17 (1972), 535–538.
[17] F. Harary, Graph Theory, Addison-Wesley. Reading, 1972.
[18] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Fundamentals of Domination in Graphs, Marcel Dekker Inc., New York, 1998.
[19] I.Z. Milovanoviic and E.I. Milovanoviic, Remarks on the energy and the minimum dominating energy of a graph, MATCH Commun. Math. Comput. Chem, 75 (2016), 305–314.
[20] K. Monius, Eigenvalues of zero-divisor graphs of finite commutative rings, J. Alg. Combin. 54 (2021), 787–802.
[21] F. Movahedi, The relation between the minimum edge dominating energy and the other energies, Discrete Math. Algorithms Appl. 12 (2020), no. 6, 2050078 (14 pages).
[22] F. Movahedi, Bounds on the minimum edge dominating energy of induced subgraphs of a graph, Discrete Math. Algorithms Appl. 13 (2021), no. 06, 2150080.
[23] F. Movahedi and M.H. Akhbari, New results on the minimum edge dominating energy of a graph, J. Math. Ext. 16 (2022), no. 5, 1–17.
[24] N. Palanivel and A.V. Chithra, Signless Laplacian energy, distance Laplacian energy and distance
signless Laplacian spectrum of unitary addition Cayley graphs, Linear Multilinear Alg. (2021), 1–22.
[25] M.R. Rajesh Kanna, B.N. Dharmendra and G. Sridhara, The minimum dominating energy of a graph, Int. J. Pure App. Math. 85 (2013), 707–718.
[26] B.S. Reddy, R. Jain and N. Laxmikanth, Eigenvalues and Wiener index of the zero-divisor graph Γ(Zn), arXivpreprint arXiv:1707.05083 (2017).
[27] M. Robbiano, R. Jimenez, Applications of a theorem by Ky Fan in the theory of Laplacian energy of graphs, MATCH Commun. Math. Comput. Chem. 62 (2009), 537–552.
[28] S. Suthar and O. Prakash, Adjacency Matrix and Energy of the Line Graph of Γ(Zn), arXiv: Combinatorics (2018), arXiv:1806.08944.
[29] B. Zhou, More on energy and Laplacian energy, MATCH Commun. Math. Comput. Chem. 64 (2010), 75–84.
Volume 14, Issue 7
July 2023
Pages 207-216
  • Receive Date: 12 March 2022
  • Revise Date: 05 August 2022
  • Accept Date: 13 September 2022