Finite solvable groups whose character graphs are trees.
For a finite group \(G\), the character graph \(\Gamma(G)\) has the nonlinear irreducible complex characters as its vertices, and two of them are joined by an edge whenever their degrees have a common prime divisor. -- This graph has been studied quite extensively in recent years. In the paper under review it is shown that for solvable groups, \(\Gamma(G)\) has no triangles if and only if \(\Gamma(G)\) has only two vertices, or \(G\) is isomorphic to the symmetric group \(S_4\). The condition of having no triangles is also equivalent to \(\Gamma(G)\) being a tree.
scientific article; zbMATH DE number 5144423
scientific article