We characterize sets of periods of cycles which arbitrary continuous graph maps may have. In this investigation, we need the spectral decomposition for graph maps [Blokh, A.M. "On dynamical systems on one-dimensional branched manifolds", Theory Funct. Funct. Anal. Appl. Kharkov 46 (1986) pp. 8-18 (Russian); 47 (1986) pp. 67-77; 48 (1987) pp. 32-46] briefly described in "The Spectral Decomposition" Section.
