We introduce a novel method, called Dispersion Entropy for Graph Signals, $DE_G$, as a powerful tool for analysing the irregularity of signals defined on graphs. We demonstrate the effectiveness of $DE_G$ in detecting changes in the dynamics of signals defined on synthetic and real-world graphs, by defining mixed processing on random geometric graphs or those exhibiting with small-world properties. Remarkably, $DE_G$ generalises the classical dispersion entropy for univariate time series, enabling its application in diverse domains such as image processing, time series analysis, and network analysis, as well as in establishing theoretical relationships (i.e., graph centrality measures, spectrum). Our results indicate that $DE_G$ effectively captures the irregularity of graph signals across various network configurations, successfully differentiating between distinct levels of randomness and connectivity. Consequently, $DE_G$ provides a comprehensive framework for entropy analysis of various data types, enabling new applications of dispersion entropy not previously feasible, and revealing relationships between graph signals and its graph topology.