On the energy of non-commuting graphs

Subject Areas : Combinatorics, Graph theory

M.  Ghorbani ¹ (Department of Mathematics, Faculty of Science, Shahid Rajaee Teacher Training University, Tehran, 16785-136, Iran)
Z.  Gharavi-Alkhansari ² (Department of Mathematics, Faculty of Science, Shahid Rajaee Teacher Training University, Tehran, 16785-136, Iran)

Received: 2017-04-01 Accepted : 2017-08-29 Published : 2017-04-01

Keywords: linear group, Non-commuting graph, Characteristic polynomial,

Abstract :

For given non-abelian group G, the non-commuting (NC)-graph $\Gamma(G)$ is agraph with the vertex set $G$\ $Z(G)$ and two distinct vertices $x, y\in V(\Gamma)$ areadjacent whenever $xy \neq yx$. The aim of this paper is to compute the spectra ofsome well-known NC-graphs.

References:

[1] A. Abdollahi, S. Akbari, H. R. Maimani, Non-commuting graph of a group, J. Algebra. 298 (2006), 468-492.

[2] M. R. Darafsheh, Groups with the same non-commuting graph, Discrete Appl. Math. 157 (2009), 833-837.

[3] M. Ghorbani, Z. Gharavi-AlKhansari, An algebraic study of non-commuting graphs, Filomat. 31 (2017), 663- 669.

[4] I. Gutman, The energy of a graph, Ber. Math. Statist. Sekt. Forschungsz Graz. 103 (1978), 1-22.

[5] A. R. Moghaddamfar, W. J. Shi, W. Zhou, A. R. Zokayi, On the non-commuting graph associated with a finite group, Siberian Math. J. 46 (2005), 325-332.

[6] G. L. Morgan, C. W. Parker, The diameter of the commuting graph of a finite group with trivial centre, J. Algebra 393 (2013), 41-59.

[7] B. H. Neumann, A problem of Paul Erd˝os on groups, J. Austral. Math. Soc. Ser. A. 21 (1976). 467-472.