The Gruenberg-Kegel graph of finite solvable groups that are character-quadratic or semi-rational
A finite group $G$ is said to be semi-rational if the set of generators of each cyclic subgroup of $G$ is contained in at most two $G$-conjugacy classes. This is equivalent to the following condition: for every column of the character table of $G$, the values appearing in the column are contained in a quadratic extension of the field of rational numbers (possibly a different one for each column). When the analogous condition holds for the rows, that is, when the field of values of every irreducible character is contained in a quadratic extension of the rationals, we say that the group is character-quadratic (these groups are often called quadratic rational in the literature). We obtain several results concerning the structure of the Gruenberg-Kegel graph of a finite solvable group that is either character-quadratic or semi-rational. More precisely, we first provide a complete classification of such graphs in the disconnected case. Also, we prove that if the graph has at most three vertices and the group is nontrivial, then it belongs to an explicit list of $20$ graphs (in the semi-rational case, this result is proved under the additional assumption that the order of the group is not divisible by $17$), and all of them are realizable except perhaps one. Finally, we show that if the graph has four vertices, then it must have at least four edges.