6533b85efe1ef96bd12bfd8a

RESEARCH PRODUCT

Bounding the number of vertices in the degree graph of a finite group

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Abstract Let G be a finite group, and let cd ( G ) denote the set of degrees of the irreducible complex characters of G . The degree graph Δ ( G ) of G is defined as the simple undirected graph whose vertex set V ( G ) consists of the prime divisors of the numbers in cd ( G ) , two distinct vertices p and q being adjacent if and only if pq divides some number in cd ( G ) . In this note, we provide an upper bound on the size of V ( G ) in terms of the clique number ω ( G ) (i.e., the maximum size of a subset of V ( G ) inducing a complete subgraph) of Δ ( G ) . Namely, we show that | V ( G ) | ≤ max { 2 ω ( G ) + 1 , 3 ω ( G ) − 4 } . Examples are given in order to show that the bound is best possible. This completes the analysis carried out in [1] where the solvable case was treated, extends the results in [3] , [4] , [9] , and answers a question posed by the first author and H.P. Tong-Viet in [4] .