Computation of Some Graph Energies of the Zero-Divisor Graph Associated‎~ ‎with the Commutative Ring $\mathbb{Z}_{p^{2}}[x]/\langle x^{2} \rangle$

Document Type : Research Paper

Authors

Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Vellore, Tamil Nadu, India

Abstract

‎Let $\mathcal{R}$ be the commutative ring $\mathcal{R}=\mathbb{Z}_{p^2}[x]/\langle x^{2} \rangle$ with identity and ${Z^{*}}(\mathcal{R})$ be the set of all non-zero zero-divisors of $\mathcal{R}$‎. ‎Then‎, ‎$\Gamma(\mathcal{R})$ is said to be a zero-divisor graph if and only if $a \cdot b= 0$ where $a,b \in V(\Gamma(\mathcal{R})) = {Z^{*}}(\mathcal{R})$ and $(a,b) \in E(\Gamma(\mathcal{R}))$‎. ‎Let $\lambda_1,\lambda_2,\dots,\lambda_n$ be the eigenvalues of the adjacency matrix‎, ‎and let $\mu_1,\mu_2,\dots,\mu_n$ be the eigenvalues of the Laplacian matrix of $\Gamma(\mathcal{R})$‎. ‎Then %the energy of $\Gamma(\mathcal{R})$ is defined as the sum of the absolute values of the eigenvalues of the graph $\Gamma(\mathcal{R})$ and the Laplacian energy of $\Gamma(\mathcal{R})$ is the sum of the absolute deviations of its Laplacian matrix's eigenvalues of the graph $\Gamma(\mathcal{R})$‎. ‎In this paper‎,
‎we discuss the energy $\mathcal{E}(\Gamma(\mathcal{R}))=\sum_{i=1}^n \abs{\lambda_{i}}$ and the Laplacian energy $\mathcal{LE}(\Gamma(\mathcal{R}))=\sum_{i=1}^n \abs{\mu_{i}-\frac{2m}{n}}$ where $n$ and $m$ are the order and size of $\Gamma(\mathcal{R})$‎.

Keywords

Main Subjects

References

[1] C. J. Rayer and R. S. Jeyaraj, Applications on topological indices of zerodivisor graph associated with commutative rings, Symmetry 15 (2023) p. 335, https://doi.org/10.3390/sym15020335.
[2] C. Johnson and R. Sankar, Graph energy and topological descriptors of zero-divisor graph associated with commutative ring, J. Appl. Math. Comput. 69 (2023) 2641–2656, https://doi.org/10.1007/s12190-023-01837-z.
[3] B. J. McClelland, Properties of the latent roots of a matrix: the estimation of $\pi$ electron energies, J. Chem. Phys. 54 (1971) 640–643, https://doi.org/10.1063/1.1674889.
[4] I. Beck, Coloring of commutative rings, J. Algebra 116 (1988) 208–226, https://doi.org/10.1016/0021-8693(88)90202-5.
[5] D. F. Anderson and P. S. Livingston, The zero-divisor graph of a commutative ring, J. Algebra 217 (1999) 434–447, https://doi.org/10.1006/jabr.1998.7840.
[6] D. F. Anderson, M. C. Axtell and J. A. Stickles, Zero-Divisor Graphs in Commutative Rings, Commutative Algebra. Springer, New York, 2011.
[7] M. R. Ahmadi and R. Jahani-Nezhad, Energy and Wiener index of zero-divisor graphs, Iranian J. Math. Chem. 2 (2011) 45–51, https://doi.org/ 10.22052/IJMC.2011.5166.
[8] B. A. Rather, M. Aijaz, F. Ali, N. Mlaiki and A. Ullah, On distance signless Laplacian eigenvalues of zero-divisor graph of commutative rings, Aims Math. 7 (2022) 12635–12649, https://doi.org/10.3934/math.2022699.
[9] P. Singh and V. K. Bhat, Adjacency matrix and Wiener index of zero-divisor graph $\Gamma(Zn)$, J. Appl. Math. Comput. 66 (2021) 717–732, https://doi.org/10.1007/s12190-020-01460-2.
[10] N. Akgunes and Y. Nacaroglu, Some proprties of zero-divisor graph obtained by the ring $Zp \times Zq \times Zr$, Asian-Eur. J. Math. 12 (2019) p. 2040001, https://doi.org/10.1142/S179355712040001X.
[11] A. N. A. Koam, A. Ahamad and A. Haider, On eccentric topological indices based on edges of zero-divisor graphs, Symmetry 11 (2019) p. 907, https://doi.org/10.3390/sym11070907.
[12] A. A. H. Ahmadini, A. N. A. Koam, A. Ahamad, M. Baca and A. Semanicová–Fenovcíková, Computing vertex-based eccentric topological descriptors of zero-divisor graph associated with commutative rings, Math. Probl. Eng. 2020 (2020) Article ID 2056902, https://doi.org/10.1155/2020/2056902.
[13] K. Elahi, A. Ahmad and R. Hasni, Construction algorithm for zero-divisor graphs of finite commutative rings and their vertex-based eccentric topological indices, Mathematics 6 (2018) p. 301, https://doi.org/10.3390/math6120301.
[14] U. Hebisch and H. J. Weinert, Semirings: Algebraic Theory and Applications in Computer Science, World Scientific Publishing, Singapore (1998).
[15] B. Hurley and T. Hurley, Group ring cryptography, Int. J. Pure Appl. Math. 69 (2011) 67–86.
[16] I. Gutman and N. Trinajstic, Graph theory and molecular orbitals. Total $\phi$- electron energy of alternant hydrocarbons, Chem. Phys. Lett. 17 (1972) 535–538, https://doi.org/10.1016/0009-2614(72)85099-1.
[17] I. Gutman, The energy of a graph, Ber. Math- Statist. Sekt. Forschungsz. Graz 103 (1978) 1–22.
[18] J. H. Koolen and V. Moulton, Maximal energy graphs, Adv. Appl. Math. 26 (2001) 47–52, https://doi.org/10.1006/aama.2000.0705.
[19] G. Caporossi, D. Cvetkovic, I. Gutman and P. Hansen, Variable neighborhood search for extremal graphs. 2. Finding graphs with extremal energy, J. Chem. Inf. Comput. Sci. 39 (1999) 984–996, https://doi.org/10.1021/ci9801419.
[20] I. Gutman and B. Zhou, Laplacian energy of a graph, Linear Algebra Appl. 414 (2006) 29–37, https://doi.org/10.1016/j.laa.2005.09.008.
[21] R. Grone and R. Merris, The laplacian spectrum of a graph II, Siam J. Discrete Math. 7 (1994) 221–229.
[22] R. Grone, R. Merris and V. S. Sunder, The laplacian spectrum of a graph, Siam J. Matrix Anal. Appl. 11 (1990) 218–238, https://doi.org/10.1137/0611016.
[23] R. Merris, Laplacian matrices of graphs: a survey, Linear Algebra Appl. 197-198 (1994) 143–176, https://doi.org/10.1016/0024-3795(94)90486-3.
[24] B. Zhou and I. Gutman, On laplacian energy of graphs, MATCH Commun. Math. Comput. Chem. 57 (2007) 211–220.
Iranian Journal of Mathematical Chemistry
Volume 15, Issue 2
June 2024
Pages 79-90

Files

Share

How to cite

Statistics