Data and models can naturally be represented by graphs. Graph representation of data is used in many areas of science and engineering, making graph matching still currently important. Besides conventional graph-matching algorithms, some successful attempts of utilizing recursive neural networks in this area have been made. In this paper, we extend previous research by proposing a novel approach using a combination of fuzzy logic and recursive neural network, which we named the fuzzy graph neural network. Adding fuzzy logic to the existing recursive neural network approach enables us to interpret graph-matching result as the similarity to the learned graph. This way, we have created a neural network, which is more resilient to the introduced input noise than a classical nonfuzzy supervised-learning-based neural network. An implementation of the proposed fuzzy graph neural network is presented in this paper. Testing of the implementation is done by using standard graph-matching datasets and problems, and includes assessment of the relation between noise and recognition accuracy for the proposed fuzzy graph neural network.