On the Energy of the line graph of Unitary Cayley graphs

Articles in Press

Document Type : Original paper

Author

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

Abstract

The energy of a graph $G$ is the sum of the absolute values of the eigenvalues of its adjacency matrix. The energy of the line graph of graph $G$ is denoted by $E(L(G))$.  The unitary Cayley graph $X_n$ is a graph with the vertex set $Z_n=\{0, 1, \ldots, n-1\}$ and the edge set $\{(a,b) \, : \, ged(a-b,n)=1\}$. In this paper, we focus on the line graph of the unitary Cayley graph $X_n$ and compute the spectrum of line graphs of $X_n$ and its complement graph $\overline{X_n}$. We also obtain the energy of the line graph of $X_n$ and $\overline{X_n}$.

Keywords

Main Subjects

References

[1] Ş.B. Bozkurt and D. Bozkurt, On incidence energy, MATCH Commun. Math. Comput. Chem 72 (2014), 215–225.
[2] B. Chen and J. Huang, On unitary Cayley graphs of matrix rings, Discrete Math. 345 (2022), no. 1, Article ID: 112671.
https://doi.org/10.1016/j.disc.2021.112671
[3] 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
[4] D. Cvetković, P. Rowlinson, and S. Simić, An Introduction to the Theory of Graph Spectra, Cambridge University Press, New York, 2009.
[5] D.M. Cvetković, M. Doob, and H. Sachs, Spectra of Graphs: Theory and Application, Academic Press, New York, 1980.
[6] 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
[7] I. Gutman, The energy of a graph, Ber. Math. Statist. Sekt. Forsch. Graz 103 (1978), 1–22.
[8] 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
[9] F. Harary, Graph Theory, Addison-Wesley Publishing Company, 1969.
[10] Y. Hou and I. Gutman, Hyperenergetic line graphs, MATCH Commun. Math. Comput. Chem 43 (2001), 29–39.
[11] 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
[12] W. Klotz and T. Sander, Some properties of unitary Cayley graphs, Electron. J. Comb. 14 (2007), Article Number: R45
https://doi.org/10.37236/963
[13] X. Li, Y. Shi, and I. Gutman, Graph Energy, Springer, New York, 2012.
[14] F. Movahedi, The energy and edge energy of some Cayley graphs on the abelian group $\mathbb{Z}_n^4$, Commun. Comb. Optim. 9 (2024), no. 1, 119–130.
https://doi.org/10.22049/cco.2023.28642.1647
[15] N. Palanivel and A.V. Chithra, Energy and Laplacian energy of unitary addition Cayley graphs, Filomat 33 (2019), no. 11, 3599–3613.
https://doi.org/10.2298/FIL1911599P
[16] H.N. Ramaswamy and C.R. Veena, On the energy of unitary Cayley graphs, Electron. J. Comb. 16 (2009), no. 1, Article Number: N24
https://doi.org/10.37236/262
Communications in Combinatorics and Optimization

Articles in Press, Accepted Manuscript
Available Online from 17 July 2024

Files

Share

How to cite

Statistics