On the eigenvalues of non-commuting graphs

Document Type : Research Paper


Department of Mathematics, Faculty of Science, Shahid Rajaee, Teacher Training University, Tehran, 16785-136, I. R. Iran



The non-commuting graph $\Gamma(G)$ of a non-abelian group $G$ with the center $Z(G)$ is a graph with the
vertex set $V(\Gamma(G))=G\setminus Z(G)$ and two distinct vertices $x$ and $y$ are adjacent in $\Gamma(G)$ if and only if $xy \neq yx$. The aim of this paper is to compute the spectra of some well-known NC-graphs.


[1] A. Abdollahi, S. Akbari, H. R. Maimani, Non-commuting graph of a group, J. Algebra 298 (2006) 468-492.
[2] A. Abdollahi, S.M.J. Amiri, A.M. Hassanabadi, Groups with speci c number of centralizers, Houston J. Math. 33(1) (2007) 43-57.
[3] O. Ahmadi, N. Alon, L. F. Blake, I. E. Shparlinski, Graphs with integral spectrum, Linear Alg. Appl. 430 (2009) 547-552.
[4] S. J. Baishya, On nite groups with speci c number of centralizers, Int. Electronic J. Algebra 13 (2013) 53-62.
[5] K. Balinska, D. Cvetkovic, Z. Rodosavljevic, S. Simic, D. A. Stevanovic, Survey on integral graphs, Univ. Beograd, Publ. Elektrotehn. Fak. Ser. Mat. 13 (2003) 42-65.
[6] S. M. Belcastro, G. J. Sherma, Counting centralizers in nite groups, Math. Magazine 67(5) (1994) 366-374.
[7] N. L. Biggs, Algebraic Graph Theory, Cambridge University Press, 1974.
[8] R. A. Brualdi, D. Cvetkovic, A Combinatorial Approach to Matrix Theory and Its Applications, Chapman and Hall/CRC; Second edition, 2008.  Vol. 4 No. 2 (2017) 27-38.
[9] F. C. Bussemaker, D. Cvetkovic, There are exactly 13 connected, cubic, integral graphs, Univ. Beograd, Publ. Elektrotehn. Fak. Ser. Mat. Fiz. 544{576 (1976) 43-48.
[10] D. Cvetkovic, Cubic integral graphs, Univ. Beograd, Publ. Elektrotehn. Fak. Ser. Mat. Fiz. 498-541 (1975) 107-113.
[11] D. Cvetkovic, P. Rowlinson, S. Simic, An introduction to the theory of graph spectra, London Mathematical
Society, London, 2010.
[12] M. R. Darafsheh, Groups with the same non-commuting graph, Discrete Appl. Math. 157 (2009) 833-837.
[13] M. Ghorbani, Z. Gharavi-Alkhansari, A note on integral non-commuting graphs, Filomat 313 (2017) 663-669.
[14] M. Ghorbani, Z. Gharavi-Alkhansari, Some properties of non-commuting graphs, submited.
[15] M. Ghorbani, F. Nowroozi-Larki, On the spectrum of nite Cayley graphs, Journal of Discrete Mathematical Sciences and Cryptography 21 (2018) 83-112.
[16] M. Ghorbani, F. Nowroozi-Larki, On the Spectrum of Cayley Graphs Related to the Finite Groups, Filomat 31 (2017) 6419-6429.
[17] M. Ghorbani, F. Nowroozi-Larki, On the spectrum of Cayley graphs, Sib. Elektron. Mat. Izv. 13 (2016) 1283-1289.
[18] F. Harary, A. J. Schwenk, Which graphs have integral spectra?, in: R. Bari, F. Harary (Eds.), Graphs and Combinatorics, Lecture Notes in Mathematics, 406, Springer, Berlin (1974) 45-51.
[19] A. R. Moghaddamfar, W. J. Shi, W. Zhou, A. R. Zokayi, On the non-commuting graph associated with a nite group, Siberian Math. J. 46 (2005) 325-332.
[20] G. L. Morgan, C. W. Parker, The diameter of the commuting graph of a nite group with trivial centre, J. Algebra 393 (2013) 41-59.
[21] B. H. Neumann, A problem of Paul Erd}os on groups, J. Austral. Math. Soc. Ser. A 21 (1976) 467-472.
[22] J. H. Smith, Some properties of the spectrum of a graph, Combinatorial Structures and their Applications,Gordon and Breach, New York (1970) 403-406