Document Type : Ischia Group Theory 2016

**Author**

Department of Mathematics, Gebze Technical University, P.O.Box 41400, Gebze, Turkey

**Abstract**

Let $G$ be a finite group. The prime degree graph of $G$, denoted by $\Delta(G)$, is an undirected graph whose vertex set is $\rho(G)$ and there is an edge between two distinct primes $p$ and $q$ if and only if $pq$ divides some irreducible character degree of $G$. In general, it seems that the prime graphs contain many edges and thus they should have many triangles, so one of the cases that would be interesting is to consider those finite groups whose prime degree graphs have a small number of triangles. In this paper we consider the case where for a nonsolvable group $G$, $\Delta(G)$ is a connected graph which has only one triangle and four vertices.

**Keywords**

**Main Subjects**

[1] B. Huppert and W. Lempken, Simple groups of order divisible by at most four primes, *Proc. of the F. Scorina Gomel State Univ.*, **16** (2000) 64–75.

[2] M. L. Lewis, An overview of graphs associated with character degrees and conjugacy class sizes in finite groups, *Rocky Mountain J. Math.*, **38** (2008) 175–211.

[3] M. L. Lewis and D. L. White, Four-vertex degree graphs of nonsolvable groups, *J. Algebra*, **378** (2013) 1–11.

[4] H. P. Tong-Viet, Groups whose prime graphs have no triangles, *Journal of Algebra*, **378** (2013) 196–206.

[5] H. P. Tong-Viet, Prime graphs of finite groups with a small number of triangles, *Monatsh. Math.*, **175** (2014) 457–484.

[6] D. L. White, Charcater degrees of extensions of PSL(2,q) and SL(2,q), *J. Group Theory*, **16** (2013) 1–33.

Proceedings of the Ischia Group Theory 2016-Part III

September 2018Pages 1-6