The massive deployment of distributed acquisition and signal processing systems, as well as the ubiquity of connected devices, is currently contributing to the development of graph signal processing. Nevertheless, this discipline still suffers from the lack of several theories and methods widely developed in signal processing. In particular, current research activities focus mainly on the processing of static graph signals (with respect to time) despite the natural anchoring in dynamic application contexts. We note, in fact, the lack of works on (on-line) identification of systems operating on streaming graph signals, and on adaptive algorithms to adapt to changes in their statistical properties over time. The objective of this paper is to introduce new tools for adaptive graph signal processing. In the first part, we propose an adaptive filtering method for streaming graph signals based on the LMS. Since this algorithm is centralized, we show how to distribute it across the graph nodes using diffusion adaptation. In the second part, we analyze the performance of the diffusion graph-LMS in both the mean and mean-square sense, as well as its stability