On rainbow vertex antimagic coloring and its application to the encryption keystream construction

Let G = (V,E) be a graph that is a simple, connected and un-directed graph. We now introduce a new notion of rainbow vertex antimagic coloring. This is a proper development of antimagic labeling with rainbow vertex coloring. The weight of a vertex v ∈ V(G) under f for f : E(G) → {1,2, …, |E(G)|} is wf (v) = S e∈E(v) f (e), where E(v) is the set of vertices incident to v. If each vertex has a different weight, afterwards the function f is also referred to as vertex antimagic edge labeling. If all internal vertices on the u−v path have different edge weights for each vertex u and v, afterwards the path is assumed to be a rainbow path. The minimum amount of colors assigned over all rainbow colorings that result from rainbow vertex antimagic labelings of G is the rainbow vertex antimagic connection number of G, rvac(G). For the purpose of trying to find some new lemmas or theorems about rvac(G), we will prove the specific value of the rainbow vertex antimagic connection number of a specific family of graphs in this paper. Furthermore, based on our obtained lemmas and theorems, we use it for constructing an encryption keystream for robust symmetric cryptography. Moreover, to test the robustness of our model, we compare it with normal symmetric cryptography such as AES and DES.

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