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On the vanishing prime graph of solvable groups. (English) Zbl 1196.20029
Let $$G$$ be a finite group and $$\text{Irr}(G)$$ the set of its irreducible complex characters. $$g\in G$$ is called ‘vanishing’ if $$\chi(g)=0$$ for some $$\chi\in\text{Irr}(G)$$. In the paper under review a new graph $$\Gamma(G)$$ is introduced as follows: The vertices of $$\Gamma(G)$$ are the prime numbers which divide the order of at least one vanishing element of $$G$$, and two such primes are connected by an edge if and only if their product divides the order of some vanishing element. For solvable $$G$$ the paper then establishes some basic properties of $$\Gamma(G)$$. For example, it has at most two connected components, and the groups with exactly two connected components can be characterized. Moreover, the diameter of $$\Gamma(G)$$ is at most 4, and this bound is best possible. Also, there is no universal upper bound on the size of an independent set of $$\Gamma(G)$$.

##### MSC:
 20D60 Arithmetic and combinatorial problems involving abstract finite groups 20C15 Ordinary representations and characters 20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, $$\pi$$-length, ranks 05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
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