The emerging field of signal processing on graphs merges algebraic or spectral graph theory with discrete signal processing techniques to process signals on graphs. In this paper, a definition of the fractional Fourier transform on graphs (GFRFT) is proposed and consolidated, which extends the discrete fractional Fourier transform (DFRFT) in the same sense the graph Fourier transform (GFT) extends the discrete Fourier transform (DFT). The definition is based on the eigenvalue decomposition method of defining DFRFT, for it satisfies all the agreeable properties expected of the discrete fractional Fourier transform. Properties of the GFRFT are discussed, and examples of GFRFT of some graph signals are given to illustrate the transform.