A graph G = (V(G), E(G) is said to permit prime distance labeling (PDL) if there exists a 1-1 labeling f: V (G) → Z such that for any two adjacent nodes u and v in G, the integer |f(u) − f(v)| is a prime. So G is a prime distance graph (PDG) if and only if there is a PDL of G. Similarly, a graph G′ = (V(G′), E(G′)) with p −nodes is said to allow prime labeling (PL) if there exists a bijection f: V(G′) → {1, 2, . . . , p} such that for each line e = uv,GCD{f(u), f(v)} = 1. A graph that admits PL is called a prime graph (PG). In this article, PDL and PL of various classes of graphs are investigated.
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