Skip to main content
Log in

Solvable groups whose prime divisor character degree graphs are 1-connected

  • Published:
Monatshefte für Mathematik Aims and scope Submit manuscript


A cut vertex of a graph is a vertex whose removal causes the resulting graph to have more connected components. We show that the prime divisor character degree graph of a solvable group has at most one cut vertex. We also prove that a solvable group whose prime divisor character degree graph has a cut vertex has at most two normal nonabelian Sylow subgroups. We determine the structures of those solvable groups whose prime divisor character degree graph has a cut vertex and has two normal nonabelian Sylow subgroups. Finally, we characterize a subgroup determined by the prime associated with the cut vertex in terms of the structure of the prime divisor character degree graph.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others


  1. Bondy, A., Murty, U.S.R.: Graph Theory. Graduate Texts in Mathematics. Springer, New York (2008)

    Google Scholar 

  2. Casolo, C., Dolfi, S., Pacifici, E., Sanus, L.: Groups whose character degree graph has diameter three. Isr. J. Math. 215, 523–558 (2016)

    Article  MathSciNet  Google Scholar 

  3. Isaacs, I.M.: Character Theory of Finite Groups. Academic Press, New York (1976)

    MATH  Google Scholar 

  4. Lewis, M.L.: Solvable groups whose degree graphs have two connected components. J. Group Theory 4, 255–275 (2001)

    Article  MathSciNet  Google Scholar 

  5. Lewis, M.L.: Bounding Fitting heights of character degree graphs. J. Algebra 242, 810–818 (2001)

    Article  MathSciNet  Google Scholar 

  6. Lewis, M.L.: Solvable groups with degree graphs having 5 vertices and diameter 3. Comm. Algebra 30, 5485–5503 (2002)

    Article  MathSciNet  Google Scholar 

  7. Lewis, M.L.: An overview of graphs associated with character degrees and conjugacy classs sizes in finite groups. Rocky Mt. J. Math. 38, 175–211 (2008)

    Article  MathSciNet  Google Scholar 

  8. Manz, O., Willems, W., Wolf, T.R.: On the number of components of a graph related to character degrees. Proc. Am. Math. Soc. 103, 31–37 (1988)

    Article  MathSciNet  Google Scholar 

  9. Manz, O., Willems, W., Wolf, T.R.: The diameter of the character degree graph. J. Reine Angew. Math. 402, 181–198 (1989)

    MathSciNet  MATH  Google Scholar 

  10. Manz, O., Wolf, T.R.: Representation of Solvable Groups. Cambridge University Press, Cambridge (1993)

    Book  Google Scholar 

  11. Morresi Zuccari, C.P.: Fitting height and diameter of character degree graphs. Comm. Algebra 41, 2869–2878 (2013)

    Article  MathSciNet  Google Scholar 

  12. Pálfy, P.: On the character degree graph of solvable groups. I. Three primes. Period. Math. Hung. 36, 61–65 (1998)

    Article  MathSciNet  Google Scholar 

  13. Sass, C.B.: Character degree graphs of solvable groups with diameter three. J. Group Theory 19, 1097–1127 (2017)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to Qingyun Meng.

Additional information

Communicated by J. S. Wilson.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work was supported by China Scholarship Council (CSC) (Grant No. 201608410231) and National Natural Science Foundation of China (NSFC) (Grant Nos. 11601121, 11771356) and Science Foundation of Henan University of Technology (Grant No. 31490036).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lewis, M.L., Meng, Q. Solvable groups whose prime divisor character degree graphs are 1-connected. Monatsh Math 190, 541–548 (2019).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


Mathematics Subject Classification