The Non-Coprime Graph of Finite Groups

Document Type : Original Scientific Paper


1 Department of Pure Mathematics, International Campus of Ferdowsi University of Mashhad, Mashhad, Iran

2 Department of Mathematics, Quchan University of Technology, Quchan, Iran



The non-coprime graph Π_G of a finite group G is a graph with the vertex
set G-{e}, where two distinct vertices u and v are adjacent if they have
non-coprime orders. In this paper, the main properties of the Cartesian and
tensor product of the non-coprime graph of two finite groups are investigated.
We also describe the non-coprime graph of some special groups including the
dihedral and semi-dihedral groups. Some open questions are also proposed.


Main Subjects

[1] J. A. Bondy and U. S. R. Murty, Graph Theory with Applications, London:
Macmillan, 1976.
[2] C. Godsil and G. F. Royle, Algebraic Graph Theory, Springer Science & Business
Media, 2001.
[3] A. Iranmanesh and A. Jafarzadeh, On the commuting graph associated with
the symmetric and alternating groups, J. Algebra Appl. 7(1) (2008) 129–146.
[4] F. Mansoori, A. Erfanian and B. Tolue, Non-coprime graph of a finite group,
AIP Conference Proceedings, 1750 (1) (2016) 050017.
[5] J. S. Williams, Prime graph components of finite groups, J. Algebra, 69 (2)
(1981) 487–513.