On the spectrum of Cayley graphs
Algebra and discrete mathematics, Tome 30 (2020) no. 2, pp. 194-206.

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The set of eigenvalues of the adjacency matrix of a graph is called the spectrum of it. In the present paper, we introduce the spectrum of Cayley graphs of order $pqr$ in terms of character table, where $p,q,r$ are prime numbers. We also, stablish some properties of Cayley graphs of non-abelian groups with a normal symmetric connected subset.
Keywords: Cayley graphs, symmetric set, semi-direct product, characteristic polynomial.
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M. Ghorbani; M. Songhor. On the spectrum of  Cayley graphs. Algebra and discrete mathematics, Tome 30 (2020) no. 2, pp. 194-206. http://geodesic.mathdoc.fr/item/ADM_2020_30_2_a3/

[1] A. R. Abdollahi, A. Loghman, “Cayley graphs isomorphic to the product of two Cayley graphs”, Ars Combin. Ars Combinatoria, 126 (2016), 301–310 | MR | Zbl

[2] L. Babai, “Spectra of Cayley Graphs”, J. Combin. Theory Series B, 27 (1979), 180–189 | DOI | MR | Zbl

[3] R. A. Brualdi, D. Cvetković, A Combinatorial Approach to Matrix Theory and Its Applications, 2nd. ed., Chapman and Hall/CRC, 2008 | MR

[4] G. Chapuy and V. Féray, A note on a Cayley graph of $\mathbb{S}_n$, arXiv: 1202.4976v2

[5] D. Cvetković, M. Doob, H. Sachs, Spectra of Graphs: Theory and Application, 3rd ed.; Academic Press, 1995 | MR

[6] M. DeVos, L. Goddyn, B. Mohar and R. Šámal, “Cayley sum graphs and eigenvalues of (3,6)-fullerenes”, J. Comb. Theory Series, 99 (2009), 358–369 | DOI | MR | Zbl

[7] P. Diaconis and M. Shahshahani, “Generating a random permutation with random transpositions”, Zeit. für Wahrsch. verw. Gebiete, 57 (1981), 159–179 | DOI | MR | Zbl

[8] N. Fox, “Spectra of Semidirect Products of Cyclic Groups”, Rose-Hulman Undergraduate Mathematics Journal, 11:2 (2010), 131–147 | Zbl

[9] M. Ghorbani, “On the eigenvalues of normal edge-transitive Cayley graphs”, Bulletin of the Iranian Mathematical Society, 1 (2014), 49–56 | MR | Zbl

[10] M. Ghorbani, F. Nowroozi Larki, “Automorphism group of groups of order $pqr$”, Algebraic Structures and Their Applications, 41 (2015), 101–107 | Zbl

[11] M. Ghorbani, F. Nowroozi Larki, “On the spectrum of Cayley graphs related to the finite groups”, Filomat, 31 (2017), 6419–6429 | DOI | MR

[12] M. Ghorbani, F. Nowroozi Larki, “On the spectrum of Cayley graphs”, Siberian Electronic Mathematical Reports, 16 (2016), 1283–1289 | MR

[13] M. Ghorbani, F. Nowroozi Larki, “On the spectrum of finite Cayley graphs”, Journal of Discrete Mathematical Sciences and Cryptography, 21 (2018), 83–112 | DOI | MR

[14] M. Ghorbani, A. Seyyed-Hadi, F. Nowroozi-Larki, “Computing the eigenvalues of graphs of order $p^2q$”, Journal of Algebraic Systems, 7 (2020), 189–203 | MR

[15] M. Ghorbani, M. Songhori, M. Rajabi-Parsa, “Normal edge-transitive Cayley graphs whose order are a product of three primes”, Italian Journal of Pure and Applied Mathematics, 39 (2018), 628–635 | MR | Zbl

[16] C. D. Godsil, G. Royle, Algebraic Graph Theory, Springer, New York, 2001 | MR | Zbl

[17] H. Hölder, “Die Gruppen der Ordnungen $p^3$, $pq^2$, $pqr$, $p^4$”, Math. Ann., 1893, 371–410 | MR

[18] G. James, M. Liebeck, Representation and characters of groups, Cambridge University Press, Cambridge, 1993 | MR

[19] R. Krakovski and B. Mohar, “Spectrum of Cayley graphs on the symmetric group generated by transpositions”, Linear Algebra Appl., 437 (2012), 1033–1039 | DOI | MR | Zbl

[20] J. Lazenby, Circulant Graphs and Their Spectra, Senior Thesis, Reed College, Portland, OR, May, 2008

[21] S. L. Lee, Y. L. Luo, B. E. Sagan, Y.-N. Yeh, “Eigenvectors and eigenvalues of some special graphs, IV multilevel circulants”, Int. J. Quant. Chem., 41 (1992), 105–116 | DOI

[22] L. Lovász, “Spectra of graphs with transitive groups”, Period. Math. Hungar., 6 (1975), 191–196 | DOI | MR

[23] B. E. Sagan, The Symmetric Group, 2nd ed., Springer, New York, 2001 | MR | Zbl

[24] J. Serre, Linear Representations of Finite Groups, Springer, New York, 1977 | MR | Zbl