### Relative g-noncommuting graph of finite groups

#### Abstract

*G*be a finite group. For a fixed element

*g*in

*G*and a given subgroup

*H*of

*G*, the relative

*g*-noncommuting graph of

*G*is a simple undirected graph whose vertex set is

*G*and two vertices

*x*and

*y*are adjacent if

*x*∈

*H*or

*y*∈

*H*and [

*x*,

*y*]≠

*g*,

*g*

^{−1}. We denote this graph by

*Γ*

_{H, G}

^{g}. In this paper, we obtain computing formulae for degree of any vertex in

*Γ*

_{H, G}

^{g}and characterize whether

*Γ*

_{H, G}

^{g}is a tree, star graph, lollipop or a complete graph together with some properties of

*Γ*

_{H, G}

^{g}involving isomorphism of graphs. We also present certain relations between the number of edges in

*Γ*

_{H, G}

^{g}and certain generalized commuting probabilities of

*G*which give some computing formulae for the number of edges in

*Γ*

_{H, G}

^{g}. Finally, we conclude this paper by deriving some bounds for the number of edges in

*Γ*

_{H, G}

^{g}.

#### Keywords

#### Full Text:

PDFDOI: http://dx.doi.org/10.5614/ejgta.2022.10.1.7

#### References

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

Z. Barati, A. Erfanian, K. Khashyarmanesh, and K. Nafar, A generalization of non-commuting graph via automorphisms of a group, Comm. Algebra 42(2014), 174–185.

M.R. Darafsheh, Groups with the same non-commuting graph, Discret. Appl. Math. 157(2009), 833–837.

A.K. Das and R.K. Nath, On generalized relative commutativity degree of a finite group, Internat. Electron. J. Algebra 7(2010), 140–151.

A.K. Das, R.K. Nath, and M.R. Pournaki, A survey on the estimation of commutativity in finite groups, Southeast Asian Bull. Math. 37(2013), 161–180.

J. Dutta, D.K. Basnet and R.K. Nath, On generalized non-commuting graph of a finite ring, Algebra Colloq. 25(2018), 149–160.

P. Dutta, J. Dutta, and R.K. Nath, On Laplacian spectrum of non-commuting graph of finite groups, Indian J. Pure Appl. Math. 49(2018), 205–216.

P. Dutta and R.K. Nath, On Laplacian energy of non-commuting graphs of finite groups, J. Linear Top. Algebra 7(2018), 121–132.

P. Erds and P. Turán, On some problems of a statistical group-theory, IV, Acta. Math. Acad. Sci. Hungar. 19(1968), 413–435.

A. Erfanian, R. Rezaei, and P. Lescot, On the relative commutativity degree of a subgroup of a finite group, Comm. Algebra 35(2007), 4183–4197.

S. Ghayekhloo, A. Erfanian, and B. Tolue, The generalised non-commuting graph of a finite group, Proc. Bulg. Acad. Sci. 67(2014), 1037–1044.

P. Hall, The classification of prime-power groups, J. Reine. Angew. Math. 182(1940), 130–141.

M. Jahandideh, M.R. Darafsheh, N.H. Sarmin, and S.M.S. Omer, Conditions on the edges and vertices of non-commuting graph, J. Tech. 74(2015), 73–76.

M. Jahandideh, M.R. Darafsheh, and N. Shirali, Computation of topological indices of non-commuting graphs, Ital. J. Pure Appl. Math. 34(2015), 299–310.

M. Jahandideh, N.H. Sarmin, and S.M.S. Omer, The topological indices of noncommuting graph of a finite group, Int. J. Pure Appl. Math. 105(2015), 27–38.

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

R.K. Nath, M. Sharma, P. Dutta, and Y. Shang, On r-noncommuting graph of finite rings, Axioms 10 (2021), 233-1–233-14.

R.K. Nath and M.K. Yadav, Some results on relative commutativity degree, Rend. Circ. Mat. Palermo 64 (2015), 229–239.

B.H. Neumann, A problem of Paul Erd$ddot{rm o}$s on groups, J. Aust. Math. Soc. 21 (1976), 467–472.

M.R. Pournaki and R. Sobhani, Probability that the commutator of two group elements is equal to a given element, J. Pure Appl. Algebra 212 (2008), 727–734.

A.R. Salemkar, F. Saeedi, and T. Karimi, The structures of isoclinism of pair of groups, Southeast Asian Bull. Math. 31 (2007), 1173–1181.

M. Sharma and R.K. Nath, Relative r-noncommuting graph of finite rings, Preprint.

M. Sharma, R.K. Nath, and Y. Shang, On g-noncommuting graph of a finite group relative to its subgroups, Mathematics 9 (2021), 3147-1–3147-13.

A.A. Talebi, On the non-commuting graphs of group D2n, Int. J. Algebra 2(2008), 957–961.

B. Tolue and A. Erfanian, Relative non-commuting graph of a finite group, J. Algebra Appl. 12 (2013), 1250157-1–1250157-12.

B. Tolue, A. Erfanian, and A. Jafarzadeh, A kind of non-commuting graph of finite groups, J. Sci. Islam. Repub. Iran 25 (2014), 379–384.

E. Vatandoost and M. Khalili, Domination number of the non-commuting graph of finite groups, Electron. J. Graph Theory Appl. 6 (2) (2018), 228–237.

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