In this paper we study the phenomenon of cospectrality in generalized line graphs and in exceptional graphs. The paper contains a table of sets of Co spectral graphs with least eigenvalue at least ?2 and at most 8 vertices together with some comments and theoretical explanations of the phenomena suggested by the table. In particular, we prove that the multiplicity of the number 0 in the spectrum of a generalized line graph L(G) is at least the number of petals of the corresponding root graph G. .