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Finite Groups Whose Common-Divisor Graph is Regular

Part of: Representation theory of groups Graph theory

Published online by Cambridge University Press:  15 February 2018

Mehdi Ghaffarzadeh ,
Mohsen Ghasemi  and
Mark L. Lewis
Mehdi Ghaffarzadeh
Affiliation:
Department of Mathematics, Khoy Branch, Islamic Azad University, Khoy, Iran (ghaffarzadeh@iaukhoy.ac.ir)
Mohsen Ghasemi
Affiliation:
Department of Mathematics, Urmia University, Urmia 57135, Iran (m.ghasemi@urmia.ac.ir)
Mark L. Lewis*
Affiliation:
Department of Mathematical Sciences, Kent State University, Kent, OH 44242, USA (lewis@math.kent.edu)
*
*Corresponding author.
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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(2f)), where f ⩾ 2 or Γ(G) is a 4-regular graph with six vertices and cd(G) = cd(Alt7) or cd(Sym7).

Keywords

Frobenius groupcharacter degreesregular graph

MSC classification

Primary: 20C15: Ordinary representations and characters
Secondary: 05C25: Graphs and abstract algebra (groups, rings, fields, etc.)
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 61 , Issue 2 , May 2018 , pp. 329 - 341
Copyright
Copyright © Edinburgh Mathematical Society 2018 

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