The paper is devoted to the study of embedded eigenvalues and of resonances of quantum graphs consisting of a compact part and a finite number of semi-infinite leads. First of all one shows that for the graphs taken into consideration the resolvent resonances are the same as the scattering resonances. Next, sufficient conditions (in terms of the matrix of coupling parameters and edge lengths) for the existence of embedded eigenvalues are given. Finally, results on the embedded eigenvalues and resonances of small perturbations of graphs with edge lengths in the compact part rationally related are proved.