Are you aware of any material the presents all (or most, or many) the properties and applications of generating functions in the context of graphs?
For example I am aware of 'Generating functionology', but that book focuses of generating functions for any discrete problem. That is why I am specifically mentioning the topic of graphs.
I found very useful material in the following documents.
Newman, structure of complex networks http://arxiv.org/abs/cond-mat/0303516
Newman, random graphs as models of networks http://www.santafe.edu/media/workingpapers/02-02-005.pdf
Callaway at al. Network Robustness and Fragility: Percolation on Random Graphs http://www.uvm.edu/~pdodds/files/papers/others/2000/callaway2000.pdf
In these documents, the MGF of the degree neighbours a neighbour is derived. Also, the size of the largest connected component is calculated and a phase transition is pinpointed when the component percolates, occupying a finite fraction of the infinite network.