In many domains (e.g. Internet of Things, neuroimaging) signals are naturally supported on graphs. These graphs usually convey information on similarity between the values taken by the signal at the corresponding vertices. An interest of using graphs is that it allows to define ad hoc operators to perform signal processing. Among them, ones of paramount importance in many tasks are translations. In this paper we are interested in defining translations on graphs using a few simple properties. Namely we propose to define translations as functions from vertices to adjacent ones, that preserve neighborhood properties of the graph. We show that our definitions, contrary to other works on the subject, match usual translations on grid graphs.
We extend the concept of stationary temporal signals to stationary graph signals. Doing so, we introduce the concept of strict sense stationary and wide sense stationary graph signals as a statistical invariance through an isometric graph translation operator. Using these definitions, we propose a spectral characterisation of WSS graph signals allowing to study stationarity using only the spectral components of a graph signal. Finally, we apply this characterisation on a synthetic graph in order to study a few important stochastic graph signals. Also, using geographic data, we study weather readings on a graph of weather stations and show evidence of stationarity in the temperature readings.