The energy and edge energy of some Cayley graphs on the abelian group $\mathbb{Z}_{n}^{4}$

Document Type : Original paper

Author

Golestan University

Abstract

Let $G=(V, E)$ be a simple graph such that $\lambda_1, \ldots, \lambda_n$ be the eigenvalues of $G$. The energy of graph $G$ is denoted by $E(G)$ and is defined as $E(G)=\sum_{i=1}^{n}|\lambda_{i}|$. The edge energy of $G$ is the energy of line graph $G$. In this paper, we investigate the energy and edge energy for two Cayley graphs on the abelian group $\mathbb{Z}_{n}^{4}$, namely, the Sudoku graph and the positional Sudoku graph. Also, we obtain graph energy and edge energy of the complement of these two graphs.

Keywords

Main Subjects


[1] L.W. Beineke, R.J. Wilson, and P.J. Cameron, Topics in Algebraic Graph Theory, Cambridge University Press, 2004.
[2] Ş.B. Bozkurt and D. Bozkurt, On incidence energy, MATCH Commun. Math. Comput. Chem. 71 (2014), no. 1, 215–225.
[3] A. Cayley, Desiderata and suggestions: No. 2. The Theory of groups: graphical representation, Amer. J. Math. 1 (1878), no. 2, 174–176.
https://doi.org/10.2307/2369306
[4] S. Chokani, F. Movahedi, and S.M. Taheri, Graph energies of zero-divisor graphs of finite commutative rings, Int. J. Nonlinear Anal. Appl. 14 (2023), no. 7, 207–216.
https://doi.org/10.22075/ijnaa.2022.7136
[5] S. Chokani, F. Movahedi, and S.M. Taheri, The minimum edge dominating energy of the Cayley graphs on some
symmetric groups, Algebr. Struct. their Appl. 10 (2023), no. 2, 15–30.
https://doi.org/10.22034/as.2023.3001
[6] D. Cvetković, M. Doob, and H. Sachs, Spectra of Graphs-Theory and Application, Academic Press, New York, 1980.
[7] D. Cvetković, P. Rowlinson, and S. Simi´c, An Introduction to the Theory of Graph Spectra, Cambridge University Press, New York, 2010.
[8] K.C. Das, S.A. Mojallal, and I. Gutman, On energy of line graphs, Linear Algebra Appl. 499 (2016), 79–89.
https://doi.org/10.1016/j.laa.2016.03.003
[9] M. Ghorbani, On the energy and Estrada index of Cayley graphs, Discrete Math. Algorithms Appl. 7 (2015), no. 1, Atricle ID: 1550005.
https://doi.org/10.1142/S1793830915500056
[10] I. Gutman, The energy of a graph, Ber. Math. Statist. Sekt. Forschungsz. Graz. 103 (1978), 1–22.
[11] I. Gutman, M. Robbiano, E.A. Martins, D.M. Cardoso, L. Medina, and O. Rojo, Energy of line graphs, Linear Algebra Appl. 433 (2010), no. 7, 1312–1323.
https://doi.org/10.1016/j.laa.2010.05.009
[12] F. Harary, Graph Theory, Addison-Wesley, 1969.
[13] A. Ilić, The energy of unitary Cayley graphs, Linear Algebra Appl. 431 (2009), no. 10, 1881–1889.
https://doi.org/10.1016/j.laa.2009.06.025
[14] W. Klotz and T. Sander, Integral Cayley graphs over abelian groups, Electron. J. Combin. 17 (2010), no. 1, ID: #R81.
https://doi.org/10.37236/353
[15] N. Palanivel and A.V. Chithra, Energy and Laplacian energy of unitary addition Cayley graphs, Filomat 33 (2019), no. 11, 3599–3613.
[16] B.R. Rakshith, On Zagreb energy and edge-Zagreb energy, Commun. Comb. Optim. 6 (2021), no. 1, 155–169.
https://doi.org/10.22049/cco.2020.26901.1160
[17] H Shooshtary and J Rodriguez, New bounds on the energy of a graph, Commun. Comb. Optim. 7 (2022), no. 1, 81–90.
https://doi.org/10.22049/cco.2021.26999.1179