Lucky edge neighborhood labeling of graphs

Let $G$ be a simple graph with vertex set $V(G)$ and edge set $E(G)$ respectively. Vertex set $V(G)$ is labeled arbitrary by positive integers and $E(e)$ denote the edge label such that it is the sum of labels of vertices incident with edge e. A lucky edge neighborhood labeling of $G$ is an assignment of positive integers to the vertices of $G $ so that edge neighborhood labelings are distinct for every edge $e$.The least integer for which a graph $G$ has a lucky edge labeling from the set ${1,2,...,k}$ is called the lucky neighborhood number and is denoted by $eta_N(G)$. In this paper, we prove that $P_n$, $C_n$, $T_{m,n}$, $S(P_n^+)$ and $S(C_n^+)$ are lucky edge neighborhood labeled graphs.