Finite Groups Whose Common-Divisor Graph is Regular. Issue 2 (15th February 2018)

• Record Type: Journal Article
• Authors: Ghaffarzadeh, Mehdi; Ghasemi, Mohsen; Lewis, Mark L.
• Is Part Of: Proceedings of the Edinburgh Mathematical Society. Volume 61:Issue 2(2018)
• Page Start: 329
• Page End: 341
• Publication Date: 2018-02-15
• Subjects: Frobenius group, -- character degrees, -- regular graph; Primary 20C15, -- Secondary 05C25; Mathematics -- Periodicals; 510
• DOI: 10.1017/S0013091517000141
• Languages: English
• ISSNs: 0013-0915

Abstract

Abstract: Let G be a finite group, and write cd ( G ) for the set of degrees of irreducible characters of G . The common-divisor graph Γ( G ) associated with G is the graph whose vertex set is cd ( G )∖{1} and there is an edge between distinct vertices a and b, if ( a, b ) > 1. In this paper we prove that if Γ( G ) is a k -regular graph for some k ⩾ 0, then for the solvable groups, either Γ( G ) is a complete graph of order k + 1 or Γ( G ) has two connected components which are complete of the same order and for the non-solvable groups, either k = 0 and cd( G ) = cd(PSL2 (2 f )), where f ⩾ 2 or Γ( G ) is a 4-regular graph with six vertices and cd( G ) = cd(Alt7 ) or cd(Sym7 ).