On Laplacian energy of non-commuting graphs of finite groups
محورهای موضوعی : Combinatorics, Graph theory
1 - Department of Mathematical Sciences, Tezpur University, Napaam-784028, Sonitpur, Assam, India
2 - Department of Mathematical Sciences, Tezpur University, Napaam-784028, Sonitpur, Assam, India
کلید واژه: Non-commuting graph, L-spectrum, Laplacian energy, finite group,
چکیده مقاله :
Let $G$ be a finite non-abelian group with center $Z(G)$. The non-commuting graph of $G$ is a simple undirected graph whose vertex set is $G\setminus Z(G)$ and two vertices $x$ and $y$ are adjacent if and only if $xy \ne yx$. In this paper, we compute Laplacian energy of the non-commuting graphs of some classes of finite non-abelian groups..
[1] A. Abdollahi, S. Akbari, H. R. Maimani, Non-commuting graph of a group, J. Algebra. 298 (2006), 468-492.
[2] N. M. M. Abreu, C. T. M. Vinagre, A. S. Bonifácioa, I. Gutman, The Laplacian energy of some Laplacian integral graph, Match. Commun. Math. Comput. Chem. 60 (2008) 447-460.
[3] M. Afkhami, M. Farrokhi, D. G. K. Khashyarmanesh, Planar, toroidal, and projective commuting and non-commuting graphs, Comm. Algebra. 43 (7) (2015), 2964-2970.
[4] S. Akbari, A. Mohammadian, H. Radjavi, P. Raja, On the diameters of commuting graphs, Linear. Algebra. Appl. 418 (2006), 161-176.
[5] A. R. Ashrafi, On finite groups with a given number of centralizers, Algebra Colloq. 7 (2) (2000), 139-146.
[6] S. M. Belcastro, G. J. Sherman, Counting centralizers in finite groups, Math. Magazine. 67 (5) (1994), 366-374.
[7] A. Castelaz, Commutativity degree of finite groups, M.A. thesis, Wake Forest University, 2010.
[8] A. K. Das, R. K. Nath, M. R. Pournaki, A survey on the estimation of commutativity in finite groups, Southeast. Asian. Bull. Math. 37 (2) (2013), 161-180.
[9] M. R. Darafsheh, H. Bigdely, A. Bahrami, M. D. Monfared, Some results on non-commuting graph of a finite group, Ital. J. Pure Appl. Math. 27 (2010), 107-118.
[10] J. Dutta, A study of finite groups in terms of their centralizers, M. Phil. thesis, North-Eastern Hill University, 2010.
[11] P. Dutta, J. Dutta, R. K. Nath, Laplacian Spectrum of non-commuting graphs of finite groups, Indian J. Pure Appl. Math. 49 (2) (2018), 205-215.
[12] J. Dutta, R. K. Nath, Finite groups whose commuting graphs are integral, Mat. Vesnik. 69 (3) (2017), 226-230.
[13] J. Dutta, R. K. Nath, Spectrum of commuting graphs of some classes of finite groups, Matematika. 33 (1) (2017), 87-95.
[14] J. Dutta, R. K. Nath, Laplacian and signless Laplacian spectrum of commuting graphs of finite groups, Khayyam J. Math. 4 (1) (2018), 77-87.
[15] P. Dutta, R. K. Nath, Various energies of commuting graphs of some super integral groups, Preprint.
[16] P. Erdös, P. Turán, On some problems of a statistical group-theory IV, Acta. Math. Acad. Sci. Hungar. 19 (1968), 413-435.
[17] R. R. Elvierayani, -. Abdussakir, Spectrum of the Laplacian matrix of non-commuting graph of dihedral group D2n, Proceeding International Conference: The 4th Green Technology Faculty of Science and Technology Islamic of University State Maulana Malik Ibrahim Malang, 2013.
[18] M. Ghorbani, Z. Gharavi-Alkhansari, On the energy of non-commuting graphs, J. Linear. Topological. Algebra. 6 (2) (2017), 135-146.
[19] I. Gutman, B. Zhou, Laplacian energy of a graph, Linear. Algebra. Appl. 414 (2006) 29-37.
[20] A. Iranmanesh, A. Jafarzadeh, Characterization of finite groups by their commuting graph, Acta. Math. Acad. Paedagogicae. Nyiregyhaziensis. 23 (1) (2007), 7-13.
[21] S. Kirkland, Constructably Laplacian integral graphs, Linear. Algebra. Appl. 423 (2007), 3-21.
[22] D. MacHale, How commutative can a non-commutative group be?, Math. Gaz. 58 (1974), 199-202.
[23] R. Merris, Degree maximal graphs are Laplacian integral, Linear. Algebra. Appl. 199 (1994), 381-389.
[24] G. L. Morgan, C. W. Parker, The diameter of the commuting graph of a finite group with trivial center, J. Algebra. 393 (1) (2013), 41-59.
[25] R. K. Nath, Commutativity degrees of finite groups; a survey, M. Phil. thesis, North-Eastern Hill University, 2008.
[26] R. K. Nath, Commutativity degree of a class of finite groups and consequences, Bull. Aust. Math. Soc. 88 (3) (2013), 448-452.
[27] R. K. Nath, Various spectra of commuting graphs of n-centralizer finite groups, Inter. J. Engin. Sci. Tech. 10 (2S) (2018), 170-172.
[28] C. Parker, The commuting graph of a soluble group, Bull. London. Math. Soc. 45 (4) (2013), 839-848.
[29] D. J. Rusin, What is the probability that two elements of a finite group commute?, Pacific. J. Math. 82 (1) (1979), 237-247.
[30] A. A. Talebi, On the non-commuting graphs of group D2n, Int. J. Algebra. 20 (2) (2008), 957-961.
[31] H. Wang, H. Hua, Note on Laplacian energy of graphs, Match. Commun. Math. Comput. Chem. 59 (2008), 373-380.
[32] B. Zhou, I. Gutman, On Laplacian energy of graphs, Match. Commun. Math. Comput. Chem. 57 (2007), 211-220.
