Explainable machine learning models have recently emerged as an important part of the research in artificial intelligence and aim at devising methods and techniques that are understandable for humans [26, 9, 18]. In this field, the use of concepts from topology and geometry has enabled developments that promise to make machine learning more easily interpretable, as required in many critical applications, where security and reliability are crucial elements. The research about group equivariant non-expansive operators (GENEOs) fits into this scientific context, offering the possibility of building small networks of operators that process the available information in a transparent and easily controllable way [5, 24, 7]. GENEOs have their roots in Topological Data Analysis and make available a mathematical theory for the approximation of observers, including their symmetries and shifting the attention from the data alone to the pairs (data, observer), seen as the main object of study. This change of perspective is justified by the fact that in many applications, the interest is not directly focused on data, but on approximating the experts' behavior in the presence of some given information [12]. It is indeed well known that different agents can react in completely different ways to the presence of the same data, and this implies that data comparison cannot be separated from the problem of understanding observers' characteristics and preferences.

In this paper we establish a bridge between Topological Data Analysis and Geometric Deep Learning, adapting the topological theory of group equivariant non-expansive operators (GENEOs) to act on the space of all graphs weighted on vertices or edges. This is done by showing how the general concept of GENEO can be used to transform graphs and to give information about their structure. This requires the introduction of the new concepts of generalized permutant and generalized permutant measure and the mathematical proof that these concepts allow us to build GENEOs between graphs. An experimental section concludes the paper, illustrating the possible use of our operators to extract information from graphs. This paper is part of a line of research devoted to developing a compositional and geometric theory of GENEOs for Geometric Deep Learning.