VERTEX-EDGE NEIGHBORHOOD PRIME LABELING OF SOME TREES

Let $G$ be a graph with vertex set $V(G)$ and edge set $E(G)$. For $ u\in V(G) $, $N_V(u) $ = $ \left\lbrace w\in V(G) / uw \in E(G) \right\rbrace $ and $N_E(u)$ = $ \lbrace e\in E(G) / e= uv,$ $ \text { for some}  v \in V(G) \rbrace $. A bijective function $ f: V(G)\cup E(G)\rightarrow \lbrace 1,2,3,\dots,$ $|V(G)$ $ \cup E(G)|\rbrace $ is said to be a vertex-edge neighborhood prime labeling, if for $ u \in V(G)$ with $deg(u)$ = 1$, $ gcd $ \left\lbrace f(w), f(uw) / w\in N_V(u) \right\rbrace = 1$ ; for $ u \in V(G)$ with $deg(u)>1$, $ gcd \left\lbrace f(w) / w\in N_V(u) \right\rbrace = 1$ and $ gcd \left\lbrace f(e) / e\in N_E(u) \right\rbrace = 1$. A graph which admits vertex-edge neighborhood prime labeling is called a vertex-edge neighborhood prime graph. In this paper we investigate vertex-edge neighborhood prime labeling for some trees namely coconut tree, double coconut tree, spider graph, olive tree, comb graph and $F(n,2)$-firecrackers.