I am not aware of current applications, but I would like to suggest the following that occurred to me:

In some networks, like social networks, we can consider a measure ( for example a new second Research Gate score), that would be another measure of the attractiveness of the nodes computed by us. We could assign these labels (these alternative RG scores for example) so that they would never repeat. They would therefore deliver a ranking of the nodes.

Now we might want to promote balance and diversity in the network, by encouraging that the linked pairs of nodes (node of rank i, node of rank j) differences of attractiveness (difference of the ranks between the nodes) goes consecutively (and only  once) through all the values of the nodes labels (ranks).

This way the network would have some similarities or "converge to" a sort of gracefull labelling. To reach this the network moderators could assign rewards to specific edges weights (values of the difference of the attractiveness between the nodes) still missing a real connection in the network.

The possible interest would be that learning might be maximized when you connect people with every possible combination of profile differences and not just reinforce connections between people with similar "rankings".

Since a social network might be relatively dense you tend to have more than one connection for each difference of attractiveness betwen the nodes. So one could promote the convergence of edges values occurrences not to 1 but to a value k for each possible difference of attractiveness. This way we would not get a graceful labelling, but something inspired by it. I don't know if there is a name for this. If not, maybe we could call it a "labelling with some grace".

Good luck.

Dear Chithra,

About graceful tree conjecture, I studied in 5 or 6 years ago and I find such ways to label graceful trees. These trees are of type n-stars trees,m rows. Of course, I labeled more trees type by graceful labeling and I need to more study for graceful labeling. But I am not to succeed give a paper or an article or ... . I think , we can use graceful labeling for crypto .

Mis. Chithra, If you be like, we work together in this topic .

Best regards,

M.Keikhaie.

Dear Chithra,

The following paper mentions a few applications towards the end, and it has lots of references.  Hope that helps.

http://sharif.edu/~eshghi/Graceful%20Graphs-Final%20Edition-89-12-15.pdf

Best wishes,

Ken Roberts

Thank you very much sir