Using deep learning on graphs I have directed graphs with each vertex has some label in "class01_01.gml" format. There are multiple such classes. I need to do multi class classification on these files using deep learning.

Note that each graph in a particular class has different set of vertices. 

Is anyone familiar how to apply deep learning in this context?

PS: I'm not that familiar in applying deep learning on graphs.
One of the most popular ways of doing classification on graphs is through graph convolutional networks. By running a convolution over the nodes of a graph, the neural net is able to learn the local neighborhoods of the graph. The seminal example is probably the paper Semi-Supervised Classification with Graph Convolutional Networks  by Kipf and Welling. https://arxiv.org/abs/1609.02907