Graph identification of different operations of vertex edge neighborhood prime

Abstract

Let $G(V(G), E(G))$ be a graph. Vertex edge neighborhood prime labeling is a function $h: V(G) \cup E(G) \rightarrow$ $\{1,2, \ldots,|V(G) \cup E(G)|\}$ with one to one correspondence and if (i) for $u \in V(G)$ with $\operatorname{deg}(u)=1, \operatorname{gcd}(h(v), h(u v): v \in$ $\left.N_V(u)\right)=1$. (ii) for $u \in V(G)$ with $\operatorname{deg}(u) \geq 2, \operatorname{gcd}\left(h(v): v \in N_V(u)\right)=1 . \operatorname{gcd}\left(h(e): e \in N_E(u)\right)=1$. A graph admits such labeling is called vertex edge neighborhood prime graph. In the present work we investigate with some families of graphs are vertex edge neighborhood prime graph.