Knowledge graphs are often used to represent structured information in a flexible and efficient manner, but their use in situated dialogue remains under-explored. This paper presents a novel conversational model for human--robot interaction that rests upon a graph-based representation of the dialogue state. The knowledge graph representing the dialogue state is continuously updated with new observations from the robot sensors, including linguistic, situated and multimodal inputs, and is further enriched by other modules, in particular for spatial understanding. The neural conversational model employed to respond to user utterances relies on a simple but effective graph-to-text mechanism that traverses the dialogue state graph and converts the traversals into a natural language form. This conversion of the state graph into text is performed using a set of parameterized functions, and the values for those parameters are optimized based on a small set of Wizard-of-Oz interactions. After this conversion, the text representation of the dialogue state graph is included as part of the prompt of a large language model used to decode the agent response. The proposed approach is empirically evaluated through a user study with a humanoid robot that acts as conversation partner to evaluate the impact of the graph-to-text mechanism on the response generation. After moving a robot along a tour of an indoor environment, participants interacted with the robot using spoken dialogue and evaluated how well the robot was able to answer questions about what the robot observed during the tour. User scores show a statistically significant improvement in the perceived factuality of the robot responses when the graph-to-text approach is employed, compared to a baseline using inputs structured as semantic triples.