  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.
