Here are some examples of papers containing results on graph labellings involving Fibonacci numbers:
'A new graph labeling with Tribonacci, Fibonacci and Triangular numbers' by F. Ignatius and S. Kaspar,
'Fibonacci graceful graphs' by D. Bange and A. Barkauskas,
'Fibonacci and super Fibonacci graceful labelings of some cycle related graphs' by S. Vaidya and U. Prajapati.
Is this the kind of thing you're looking for? A lot of other results are mentioned in the survey paper 'A dynamic survey of graph labeling' by Joseph Gallian.