I'm new to the area, and we don't have a course on graph neural networks at our university. However, I will still like to know the main theoretical results when considering convergence of graph neural networks. Where can I learn more about the topic? Thanks.
More generally, this area is known as geometric deep learning, i.e., it encompasses learning not only on graph but on other non-Euclidean domains. A good broad start is the survey of Bronstein et al. , after which I can recommend basically any material by Bronstein like his many excellent presentations (also available on Youtube - just do a search).
For graph neural nets, you might start with Kipf & Welling (the survey  will give references). Perhaps loosely building on their work, there is also GraphSAGE  and later developments like graph attention networks. I've not actively followed on the literature for the past 2-3 years, so I don't know what's hot currently or how badly outdated these methods might be, but these works definitely will give you the foundation.
 Bronstein, Michael M., Joan Bruna, Yann LeCun, Arthur Szlam, and Pierre Vandergheynst. "Geometric deep learning: going beyond Euclidean data." IEEE Signal Processing Magazine 34, no. 4 (2017): 18-42.
 Hamilton, William L., Rex Ying, and Jure Leskovec. "Inductive representation learning on large graphs." In Proceedings of the 31st International Conference on Neural Information Processing Systems, pp. 1025-1035. 2017.