Summary: Given any finite connected graph Gamma with n vertices, we construct a family of spider graphs of Gamma k of Gamma by attaching two-valent trees (line graphs) to the vertices of Gamma. These spider graphs have in common that all but n of their vertices have valency at most 2. When a Gamma k is large enough, most of its eigenvalues have the form zeta + zeta--1, where zeta is a root of unity. Furthermore, the largest eigenvalue lambda of Gammak is bounded above by the largest valency that occurs in Gamma, hence independent of k. We prove that, with the exception of Dynkin diagrams, there is an effective bound N such that if k > N in an partial ordering, then the number field Q(lambda2) is not an abelian extension of Q. In less vigorous terms, we could say that Q(lambda2) is not abelian when the "legs" of the spider graph Gammak are long enough.