Abstract The prime character degree graph Δ ( G ) of a finite group G is the graph whose vertices are all prime divisors of the degrees of the complex irreducible characters of G, with distinct primes p and q joined by an edge if pq divides χ ( 1 ) for some complex irreducible character χ of G. Lewis and White determined all graphs with four vertices that occur as Δ ( G ) for some nonsolvable group G. In this paper, we determine all graphs with five vertices- up to two exceptions that occur as Δ ( G ) for some nonsolvable group G. Along with previously known results on prime character degree graphs of solvable groups, this completes the classification of all five-vertex graphs- up to two exceptions that occur as Δ ( G ) for some finite group G.

中文翻译：

---

不可解群的五顶点度图

摘要 有限群 G 的素字符度图 Δ ( G ) 是这样的图，其顶点都是 G 的复不可约字符的度的素因数，如果 pq 整除 χ ( 1 ) 对于 G 的一些复杂的不可约特征 χ。 Lewis 和 White 确定了所有具有四个顶点的图，这些图出现为某些不可解群 G 的 Δ ( G )。在本文中，我们确定了所有具有五个顶点的图 - 最多两个例外对于某些不可解群 G，作为 Δ ( G ) 出现。连同可解群的素数特征度图上的先前已知结果，这完成了所有五顶点图的分类——最多两个例外，对于某些以 Δ ( G ) 发生的有限群 G。