# On the eigenvalues of Cayley graphs on generalized dihedral groups

Document Type : Research Paper

Authors

1 ‎Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj‎, ‎Iran.

2 Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran

10.29252/as.2019.1481

Abstract

‎Let $\Gamma$ be a graph with adjacency eigenvalues $\lambda_1\leq\lambda_2\leq\ldots\leq\lambda_n$‎. ‎Then the energy of‎ ‎$\Gamma$‎, ‎a concept defined in 1978 by Gutman‎, ‎is defined as $\mathcal{E}(G)=\sum_{i=1}^n|\lambda_i|$‎. ‎Also‎ ‎the Estrada index of $\Gamma$‎, ‎which is defined in 2000 by Ernesto Estrada‎, ‎is defined as $EE(\Gamma)=\sum_{i=1}^ne^{\lambda_i}$‎.
‎In this paper‎, ‎we compute the eigenvalues‎, ‎energy and Estrada index of Cayley graphs on generalized dihedral groups‎. ‎As an application‎, ‎we‎ ‎compute these items for honeycomb toroidal graphs and Cayley graphs on dihedral groups‎.

Keywords

#### References

[1] B. Alspach and M. Dean, Honeycomb toroidal graphs are Cayley graphs, Information Processing Letters, Vol. 109 (2009), pp. 705–708.
[2] M. Arezoomand and B. Taeri, On the characteristic polynomial of n-Cayley digraphs, Electron. J. Combin. Vol. 20 No.3 (2013), P57, pp. 1–14.
[3] M. Arezoomand and B. Taeri, Normality of 2-Cayley digraphs, Discrete Math., Vol. 338 (2015), pp. 41–47.
[4] L. Babai, Spectra of Cayley graphs, J. Combin. Theory Ser. B, Vol. 27 (1979), pp. 180–189.
[5] P. Diaconis and M. Shahshahani, Generating a random permutation with random transpositions, Z. Wahr. Verw. Gebiete, Vol. 57 (1981), pp. 159–179.
[6] E. Estrada, Characterization of 3D molecular structure, Chem. Phys. Lett. Vol. 319 (2000), pp. 713–718.
[7] M. Ghorbani, On the energy and Estrada index of Cayley graphs, Discrete Mathematics, Algorithms and Applications, Vol. 7 No. 1 (2015) , 155005.
[8] I. Gutman, The energy of a graph, Ber. Math. Statist. Sekt. Forschungsz. Graz Vol. 103 (1978), pp. 1–22.
[9] M. Isaacs, Character theory of finite groups, Academic Press (1976).
[10] H. Kurzwil and B. Stellmacher, The theory of finite groups, An introduction, Springer-Verlag (2004).
[11] L. Lovász, Spectra of graphs with transitive groups, Period. Math. Hungar. Vol. 6 (1975), pp. 191–196.
[12] M. J. de Resmini, and D. Jungnickel, Strongly regular semi-Cayley graphs, J. Algebraic Combin., Vol. 1 (1992), pp. 217–228.
[13] G. Sabidussi. Vertex-transitive graphs Monatsh. Math., Vol. 68 (1964), pp. 426–438.