Edge irregular reflexive labeling for the r-th power of the path

Let G(V, E) be a graph, where V(G) is the vertex set and E(G) is the edge set. Let k be a natural number, a total k-labeling phi : V(G) boolean OR E(G) -> {0, 1, 2, 3, ..., k} is called an edge irregular reflexive k-labeling if the vertices of G are labeled with the set of even numbers from {0, 1, 2, 3, ..., k} and the edges of G are labeled with numbers from {1, 2, 3, ..., k} in such a way for every two different edges xy and x'y' their weights phi(x) + phi(xy) + phi(y) and phi(x') + phi(x'y') +phi(y') are distinct. The reflexive edge strength of G, res(G), is defined as the minimum k for which G has an edge irregular reflexive k-labeling. In this paper, we determine the exact value of the reflexive edge strength for the r-th power of the path P-n, where r >= 2, n >= r + 4.