One of applications of infinite Graph theory is about boiling points of infinite symmetric graphs in Nanotechnology .
Wiener showed that the Wiener index number is closely correlated with the boiling points of Alkane molecules
see Wiener, H. J. "Structural Determination of Paraffin Boiling Points." J. Amer. Chem. Soc. 69, 17-20, 1947.
For example for the case of Carbon Nanocone, which is infinite symmetric graph 
In fact Hyper-Wiener index is topological invariant and we calculated(when I was Bachelor degree) the hyper-Wiener index of the infinite one-pentagonal Carbon Nanocone. See