The key-models and their lock-models for designing new labellings of networks

The security of real networks is facing serious challenges and is harder than ever. We propose an idea of “topological structure + number theory” for designing new passwords. By means of graph theory we define a twin odd-graceful tree T = T₁ o T₂ obtained by identifying a certain vertex of T₁ with a certain vertex of T₂ together. We prove that some particular trees (key-models) can have their associated trees (lock-models) under the concept of twin odd-graceful trees. Trees having set-ordered odd-graceful labellings T₁ with their associated trees T₂ can produce certain twin odd-graceful trees such that these twin odd-graceful trees are odd-graceful.