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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

M. R. Darafsheh, M. H. Khalifeh and H. Jolany, The Hyper-Wiener Index of One-pentagonal Carbon Nanocone, Current Nanoscience, 2013, 9, 557-560 557

We also computed (when I was Bachelor degree)the Wiener index Dendrimer Nanostar which is infinite graph

$$W_{G_n} = 972n4^n−14584^n+17552^n−270$$

M. H. Khalifeh, M. R. Darafsheh, and H. Jolany, The Wiener, Szeged, and PI Indices of a Dendrimer Nanostar, Journal of Computational and Theoretical Nanoscience Vol. 8, 220–223, 2011