P- adic distance labeling of graphs

Abstract

In this paper, we introduce a new type of Labeling of a graph G (V, E) with |V(G)| vertices and |E(G)| edges as an injective map φ: V(G) → {1, … . k}, k ∊ ℕ such that the induced constant mapping φ*: E(G) → (0, 1) by φ*(e) = φ*(uv) = |φ(u) - φ(v)|p = λ, where λ ∊ (0, 1) called p-adic distance labeling, for any fixed prime ‘p’. Here, we prove the existence of this new p−adic distance labeling for some simple graphs such as Path (Pn), Cycle (Cn), Complete graph (Kn) and Star graph (K1, n), n ≥ 3 also for the Corona graph Pn⊙k2¯, when n≥2 and even and for p=2,3. And we explore the applications of this labeling in wireless networks.