The properties of many nanoscale devices are sensitive to local atomic configurations, and so elemental identification and electronic state analysis at the scale of individual atoms is becoming increasingly important. For example, graphene is regarded as a promising candidate for future devices, and the electronic properties of nanodevices constructed from this material are in large part governed by the edge structures1. The atomic configurations at graphene boundaries have been investigated by transmission electron microscopy and scanning tunnelling microscopy2, 3, 4, but the electronic properties of these edge states have not yet been determined with atomic resolution. Whereas simple elemental analysis at the level of single atoms can now be achieved by means of annular dark field imaging5 or electron energy-loss spectroscopy6, 7, obtaining fine-structure spectroscopic information about individual light atoms such as those of carbon has been hampered by a combination of extremely weak signals and specimen damage by the electron beam. Here we overcome these difficulties to demonstrate site-specific single-atom spectroscopy at a graphene boundary, enabling direct investigation of the electronic and bonding structures of the edge atoms—in particular, discrimination of single-, double- and triple-coordinated carbon atoms is achieved with atomic resolution. By demonstrating how rich chemical information can be obtained from single atoms through energy-loss near-edge fine-structure analysis8, our results should open the way to exploring the local electronic structures of various nanodevices and individual molecules.