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Volume 45 Number 5 Volume 45 Number 6

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Research articles

ScienceAsia 45 (2019): 482-487 |doi: 10.2306/scienceasia1513-1874.2019.45.482

Some results on the non-commuting graph of a finite group

K. Moradipoura,*, Sh. Ilangovanb, S. Rashidc

ABSTRACT:     Let G be a metacyclic p-group, and let Z(G) be its center. The non-commuting graph ΓG of a metacyclic pgroup G is defined as the graph whose vertex set is G−Z(G), and two distinct vertices x and y are connected by an edge if and only if the commutator of x and y is not the identity. In this paper, we give some graph theoretical properties of the non-commuting graph ΓG. Particularly, we investigate planarity, completeness, clique number and chromatic number of such graph. Also, we prove that if G1 and G2 are isoclinic metacyclic p-groups, then their associated graphs are isomorphic.

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a Department of Mathematics, Faculty of Khorramabad, Lorestan Branch, Technical and Vocational University, Iran
b The University of Nottingham Malaysia Campus Jalan Broga, 43500 Semenyih Selangor Darul Ehsan, Malaysia
c Department of Mathematics, College of Basic Sciences, Yadegar-e-Imam Khomeini (RAH) Branch, Islamic Azad University, Tehran, Iran

* Corresponding author, E-mail: kayvanmrp@yahoo.com

Received 14 Apr 2019, Accepted 20 Aug 2019